OP in your first post at top of thread:
"We separated a few months ago but still live in the same house while he decides what happens next"
and from your later post:
"He said he needed to wait 4 - 6 months to see how he felt, but there was a chance he'd take me back."
Why are you giving him all this power?
He wants the 6 months to see how he feels about the other woman and if he can make a go of a proper relationship with her. If he had only just met the woman when he started musing the idea about an "open" marriage it was probably too soon for him to decide whether to abandon a long-standing marriage for a fleeting dalliance and very nascent feelings for another woman.
He wanted your OK to an "open" marriage so that he could have sex outside the marriage without guilt. You didn't give your permission to that ...... which is why he's now lying at every opportunity and pretending he hasn't got a bit on the side already.
His words are all wrong. If there had been a mutual decision to semi separate his words would have been more along the lines of:
"I wanted 6 months / think we both need 6 months to cool off and reflect ..... but I did see a chance that we might reconcile"
As for .......
"there was a chance he'd take me back."
Why? What had you done? Robbed a bank?
Been unfaithful to him ????
His wording is giving him all the power and actually putting the blame onto you as if you had done something wrong - the only thing you did was decline an open marriage.
He has another side piece lined up already and he's only taking this time living separate lives under the same roof as you to see if he can make a go of things with the Other Woman.
If for some reason the current OW falls thru (she becomes unavailable / doesn't want him / has a husband herself who she decides to stay with) ..... then he will shilly shall back to you (slightly) but it won't be in any kind of massive declaration of love ..... at that point (if current OW falls thru) he might want to discuss the open marriage option again so that he can continue to "fish" for extra crumpet outside the marriage.
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Sadly, the only power you have is to call him out on all his bullshit (lies) and force things to end sooner / kick him out / start divorce proceedings etc. **
** That option obviously has financial consequences and it may suit you as much as him to keep one set of household bills for 6 months until you both face the inevitable split and divvy up of marital assets.
See my other posts above re sending him a sarky "accidental" goodbye cheating scumbag loser type card.
Also try (if possible) to find some evidence about this other woman. I know it can be difficult if he has a pin lock on his phone and if he only sees this OW at (say) their mutual place of work - but doing this will also give you some power back:
(a) to be able to put all the pieces of the jigsaw together;
(b) to place you in a better position to call him out on his bullshit; and
(c) if necessary to sue him for adultery in the divorce proceedings.