For me it comes down to whether "Until the regroup phase," was intended to mean the entire "each minion skirmishing a

companion loses fierce and cannot gain fierce" part should be in effect for that period of time, or whether it was just intended to describe how long the "loses fierce and cannot gain fierce" part was supposed to last.
I'm thinking it's the latter (a 'no' to your question) because I just feel it's odd the designers would make it something like
Mount Doom. But as written, I think you've already made a case for minions at the next site losing fierce.