I think false R can mean different things... It could be that they're still seeing the OP, still in the A.
It could mean that they're not seeing the AP but that they're keeping the A alive in their mind - fantasizing, or pining, or refusing to give it up mentally. False R because then the WS maintains a commitment to the A, and that's not acceptable if you're supposed to be loyal to your spouse and M.
It could mean TT - that the WS is withholding details or even multiple As intentionally. False R for sure, because you don't know everything you're reconciling from, and because there cannot be any more secrets.
It could mean - as was in our case - continuing to see the A(s) in a positive light, not feeling at all bad, still smiling over memories of the As, and being unwilling to hear the BS's pain, getting outright annoyed - rugsweeping. Definitely false R, because the WS is not helping or taking seriously the BS's need to heal, or for that matter the WS isn't doing the work to make sure they won't make those choices again.
It could also mean going through the motions with R, but not really into it or trying half-heartedly (ETA: I think this is different than having a hard time or feeling doubts - of course we struggle in R - I mean this more about not really caring from the start if the M is saved but claiming otherwise), and in the end seeking separation or divorce. R must be very hard, but it's something both spouses must actively choose to do if it is to be R, imo. That doesn't mean everyone has to R. The A might certainly be a dealbreaker, and that's more than understandable. Also, it does not mean you must choose immediately to R - you can be honest and say you're in limbo, and while it sucks being there, it's reality. imo saying you're in R is serious. It's false R when a person says they commit to R but doesn't really put their heart in it.
I think false R can mean different things, and it can hurt in different ways. In general: I think it means when the commitment of one or both spouses is absent from the M post-A, while one or both spouses claim it's there.