Behm and 11 other people were injured when the Cowboys' practice bubble collapsed during a severe thunderstorm May 2.
Branson asked a Dallas County court late last month to order a pre-litigation deposition of Nathan Stobbe, president of facility builder Summit Structures, and its parent, Cover-All Building Systems.
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Court records show that the attorney also asked for all of the Canada-based company's documents related to the tent-like facility.
Branson told Dallas Morning News reporter Brooks Egerton that Stobbe had already submitted to questioning voluntarily. Branson said he had received some documents, too.
"The defendants at this time seem to be cooperating," he said. "I'd like to get this case up and running as smoothly as possible for everybody's sake."