Bluesea, this is a very dangerous time as a LBS. This board is littered with LBSs that were overanxious when their S started to appear to be turning around, only to see them run the other way.
So the answer is you double-down on your DB efforts. He had a lot of work to do. The worst thing you can go is to just let him back without doing the work he needs to do.If you just let him to waltz back you've taught him "hey, I can go have a PA anytime I want, she will just let me right back."
So yes, let him be the one to start conversation. Be pleasant, polite, upbeat. But be busy. Be the one to end them.
Continue to GAL. Continue to work on your self-improvements. Continue to be healthily detached (google: self-differentiation in marrage).
Also sit down alone and talk about what it will take to rebuild trust. IC for him. IC for you. MC for both of you. Full transparency from him.
Your husband has lied and cheated on you. Remember that. That is not something to just brush under a rug and forget about.
M(53), W(54),D(19) M-23, T-25 Bomb Drop - Dec.23, 2017 Ring and Piecing since March 2018