Here's my bit:
IANAL, but I'll try to give a recap as best I can.
The whole thing started off with one of the judges asking how to pronounce David Miscavige's name.
First Wallace Jefferson stood up to talk. He started off laying the general stuff: this is a dispute between the Church of Scientology and Marty and Monique Rathbun that has been going on for years. Originally it was Marty who started a campaign against the Church by posting up his blog, etc. Monique made herself a part of the conflict between Scientology and Marty by posting things on Facebook. He brought everything up to date, saying that Monique has been seeking a deposition from Miscavige since the beginning. Then reiterated the idea that CSI is the responsible party and if it were found to have committed wrong-doing, which he assures the court won't be the case, that CSI has the means and resources to make good on such a judgement. That in light of the fact that David Miscavige didn't have a role in the activities and that he has never been to Texas to perform any of these activities, this becomes Monique seeking an apex deposition. Now, he says, Monique and her legal team have had the chance to depose many members of RTC and CSI and that in none of their depositions did they ask pointed questions about David Miscavige's involvement or lack thereof in these activities taking place in Texas. This shows a lack of good faith on there part, he says, and that Miscavige shouldn't be pulled away from his "ecclesiastical mission" just to satisfy their desire to depose him. Then he said that a Texas court has no jurisdiction over David Miscavige and that if Monique wants to sue him she needs to do in California and there will be no argument over jurisdiction.
One of the judges (the female one and forgive me I know none of their names) said that she thought one of the primary questions here was whether or not by submitting to a deposition, is the person really also submitting to jurisdiction? Wallace Jefferson didn't seem to have a good answer for that, because that's when he turned the argument into the idea that Monique's team had ample opportunity to question intermediaries about Miscavige's activities or lack thereof in relation to this case, but they did not ask those questions. he says they did not ask those questions because they didn't want a negative answer. Another judge then asked, would it change things if--hypothetically--all other defendants were dropped from the lawsuit and Miscavige was the sole defendant. I'm not sure why he asked that or what the implication was--it seemed like a strange question to my untrained ear. But Wallace's response was again, no it wouldn't change things. That's when he said in that case Monique should sue David Miscavige in California. Finally the same judge asked did Waldrip get to review all of the depositions, and Wallace seemed to get oddly defensive at the question. Answering (and obviously all of this is paraphrased to the best of my ability) that no, but even if he did it would still show that Miscavige should not be deposed and that Monique failed to show food fail in her questioning of the intermediaries being deposed.
Leslie stepped up and started her argument out by saying that in the history of TX law no court has ever applied the apex deposition rule to a named defendant in a lawsuit. Then she asked how could anyone else know David Miscavige's actions with relation to this case better than Miscavige himself. She then said, if David Miscavige had come to Texas himself and taken pictures himself and so on, that they wouldn't be in the appeals court at all and there'd be no question about deposing Dear Leader--I mean Shorty McFistToCuffs. Instead he used agents to do the work for him and now there is an argument over jurisdiction. Everyone in the court room was calling that the "agency argument". She said that basically, whether Misavige had done the activities or he used someone else to do it, it doesn't change the fact that he should be deposed for his own actions. One of the judges then asked if she asked the intermediaries being deposed about David Miscavige's actions and additionally, if they didn't have the information she was looking for, then did she ask who she should be asking? Leslie then brought up the point about Warren McShane admitting to knowing little about what Miscavige was doing and pointed out that David Miscavige's office is not staffed and no one really knows what he's up to at all--except for Miscavige of course.
One of the judge's then asks about her theory, saying that is she assuming just because of Miscavige's authority, is Monique's team assuming that he knows what is going on? Leslie's response was that they have evidence which indicates that Miscavige was aware of what was going on. She also pointed out that in prior cases the appeals court did not question whether or not the trial judge had ruled correctly when he ordered a deposition. As it should be in this case, the court should go into this assuming that Waldrip's ruling that Miscavige should be deposed was correct in the first place.
After that Leslie went into the discovery process and basically said that despite their extreme efforts to pursue written discovery that the defendants failed to provide any documentation at all in, in fact, claimed it did not exist. Which Leslie said that she and her team have ample evidence that contraindicates that claim. That Scientology does keep detailed records and reams of paper related to all operations, so the evidence must exist somewhere. She then went on to say that they had requested the deposition of Miscavige after Scientology's team was uncooperative in discovery, basically being left with the idea that no one knows anything except for Miscavige. One of the judges returned to the apex idea, asking if she thought that Waldrip in his ruling was trying to follow the apex deposition rule. Leslie responded that even though Waldrip did not consider this an apex deposition that he felt that the rule would have been satisfied anyway. Then one of the judges asked again, why Leslie thought Waldrip didn't review all of the intermediary depositions before ruling. Leslie basically implied that he didn't need to because there had been ample information shared over the months that this trial has been going on. Then when it came down to the wire the proceedings were interrupted by an anti-SLAPP motion which took precedence because of the time constraint, and that was why Waldrip didn't have time to review all the intermediary depositions. She then said that she thought he had read all of the depositions, but hadn't had time to sort through Scientology's numerous objections. Which then made clear to my why Wallace had acted defensive about the question when he was asked--he knew it was his team's fault that Waldrip made a decision.
Then one of the judge's asked Leslie about the directed tort rule, and the alter ego rule. Do these apply to the case. That's when Leslie emphasized the "agency argument"--that Miscavige had used agents to act on his behalf in Texas which is basically the same thing as if he had carried out the actions himself. Then one of the judges asked about Leslie's assertion that DM was reaping personal benefit from these activities. Leslie answered by saying that she felt it was more of a psychological benefit in the satisfaction he felt from his vindictive activities, she wasn't trying to say that he was receiving monetary benefit.
Wallace came back up and complained that Monique's team keeps changing their argument that they were "all over the map" with their jurisdiction argument: first they said it was general jurisdiction because Miscavige had been in Dallas for the opening of a church; then it was alter ego; now they are arguing jurisdiction by agency. Then he repeated his argument that Monique's team had the opportunity to question intermediaries about Miscavige's involvement in other depositions, but they failed to do so. Then one of the judges asked how Wallace would direct the court in their review of the case and Wallace replied, "Abuse of discretion." Then said that the court should deny the deposition because Monique's request to depose Micavige was not in good faith. He also got in a bit about how there were declarations from former church officials that Monique's team relied on to support their request to depose Miscavige, but that those declarations were out of date and that the church was entirely different than it was back in the days of those people. That the church has taken a whole new turn and that those people in those declarations have no idea what is going on today.
The whole thing was 45 minutes. I couldn't tell whether or not the judges were leaning one way or the other by the time they left. I also thought they'd have more questions. I felt like the judges had a lot more questions for Leslie than they did Wallace.
There's the best I could do. I'm sure Tony will have input from others, and I'm sure someone will correct me if I'm wrong here! It's tough to decipher some things with my untrained ears.