I am not sure if they would ripen ok after being in the fridge. I would probably let them all get pretty much ripe on the counter and then put them in the fridge. I would think they would at least last a few days like that before the quality started dropping.

This came from the grower himself when I got mine.
He specifically instructed me to refrigerate most of them just for that purpose. I put mine in the drawers at the very bottom so it doesn't get too cold.
Taking out one or two every other day for ripening.
So far, that's worked out well for me. Maybe my pallets are not as sophisticated compared to yours but the cherimoyas were good and tasted about the same.

I've also tried the other way but made a mistake by not letting it fully ripened. When I stuck it in the fridge, the process stalled and ended up with nearly a quarter of that fruit still somewhat hard. So you gotta be careful with your inspection. hope that helps

Tim

Cherimoyas ripen faster when they are together. If you single them out and keep them away from bananas or other fruit they will last longer. Also, do not store then near paper or in paper bags -this causes them to ripen faster than usual(four or less days)

Every method mentioned if done opposite will help ripen the fruit faster.
Hope this was helpful,

Daniel