Sorry for your relationship difficulties, HikingOut.
False R is when the wayward continues the affair while working on the relationship with the BS. So I guess a BS can't do false R. Some people complain about BS who are faking R, that they are planning to leave the marriage at a later date but don't let on to the WS that this is going to happen. Some don't like this, others don't have a problem with it.
BS are not perfect just because they are BS. HO herself had an A that she thought was an exit A. I guess she was going to have the A, then file, instead of just leaving? Or use the A as an excuse for dialogue, let the A break up the marriage? Until she changed her mind?
My perspective in "BS has to play by the rules" is a bit skewed because I got left for AP. So I played by the rules, and what did it get me? I did my best in my M, but I'm just some divorced guy today.
I don't want to hijack here. I hate this saying, but WS heals WS, BS heals BS, together they heal the marriage. Well, the WS path and BS path are radically different. Plus, the WS always knows everything, knows what they did, etc. and the BS is just reacting in real time to what they find out from the WS. They're completely different scenarios and require completely different skill sets. And who knows if the BS has a skillset set up for what they will face. Whereas the WS at least has it all in front of them.
I am not blaming anyone here, just explaining that the two parties in the R face radically different tasks, and their journeys can drive them apart as easily as together. FWIW, HO's H seemed like he might have issues with stuffing his feelings. I don't know, don't really know him. But with any BS, he's got "WS cheated on me, but now they want R, they seem sincere, but I still might be angry, but they're being nice now, giving me attention, laying on the sex, I've got all of this history, would lose money, time with the kids, but they cheated, now I look at other people and wonder more..." etc. etc. etc.
So a BS is there, typically less happy than the WS is with R, and they get a potential A situation, think that maybe this situation, this person will provide something that was lost in the marriage due to the A. Or they just want to hedge their bets, aren't happy at the time with how R is going, maybe they think WS will cheat again and it's a hedge if they cheat first, or they think of it as an exit A, the A will break up the M that they don't like and they don't want to talk directly about it. Until they change their mind.
I am just trying to provide something of value here and don't want to hijack. HO has had a different life path from me and some of what I think probably doesn't apply. I have to try.