Waiting on a TS PR and have another job offer

So, long story short, I’ve been unemployed for 10 months (had 8 signed offers and positions pulled or paused, crossover completed the same day client decided not to fill position, blah blah blah). My most recent signed offer requested a crossover and the client required a periodic re-investigation even though my last one was only 6 years ago. Fine. I filled out the paperwork prior to the shutdown. I was supposed to start 10/1, but here I sit.

I’ve continued to interview because after the last 10 damn months, I’m not taking anything for granted.

IF one of the companies I’m interviewing with offers me a better position, can the PR be picked up by them? Or am I stuck waiting for the slower-than-weight-loss PR to conclude?