Stays are not mandatory. They can be requested.
Different venues, different remedies.
In VHC's case(s), the primary district court litigation against Apple has not been stayed. There have been thousands of filings and motions and requests for segregation and etc. etc. etc. Just this week, Apple asked Judge Davis (Tyler, TX) to rule that their alleged infringement (both pre- and post- complaint) was not willful... so that VHC can't ask for treble damages in the future.
So yeah, I follow that company as closely as I follow IDCC. And no portion of the main case, especially the Apple portion, has been stayed.
EDITED: OldDog alerted me to a newer smaller district case for VHC's '181 patent that was stayed when they made the parallel ITC complaint. The main district case ('135, '759, '180, '504, '151, and '211 patents) has not been stayed, still proceeding toward a scheduled November trial).