"Cuozzo merely affirmed the PTAB's use of the BRI standard and did not change anything pertaining to the motions already filed or the judge's previous denial of a stay on this issue."
As Apple pointed out in their renewed motion for stay - Virnetx argued, in its reply to the first motion, that it was likely the SCOTUS would overturn BRI in Cuozzo so the court should deny the stay because the use of BRI is up in the air and if it is overturned all of the IPR's would be sent back. Considering the above and that the Cuozzo decision wasn't handed down till June 20th I don't blame Apple for renewing the motion and it being outside HJS suggested dates for motions.
The Ultratec case cited in the motion shows that District Court can entertain and grant a stay after a trial has been completed and before all motions have been ruled on and final judgement issued - especially if the IPR's/re-exams are well advanced and not in agreement with the verdict.
The Ultratec ruling on the exact motion cited by Apple is linked below - it is 12 pages and a quick read.
I think it is unlikely that HJS grants a stay - but it certainly isn't impossible nor a slam dunk that he won't. The Ultratec damages were $44M....here we are talking about one of the largest verdicts ever at $625M. Considering the state of the patents in IPR and re-exams it does not seem like a stretch to think a District Court judge may punt to the PTAB and let them and the CAFC figure it out - just like the judge did in Ultratec in a far less important case.
Again - I'm not saying I think HJS will grant a stay...I just don't dismiss it out of hand like some tend to do here. Read the Ultratec ruling.