Will not poop in potty

My son is 3 years and 4 months. He has been pee potty trained for months, wears undies during the day. He only uses a diaper at night and during nap time at daycare (he doesn’t nap at home).

However, he has never once pooped on the potty. We’ve tried to be patient, he hates the feeling of the poop on him while we clean him up but he refuses to try to poop on the potty. On the occasion we do get him to try it feels like either he didn’t really need to and he eventually is just done OR he feels it coming and gets scared and wants to get up and then poops like five minutes later in his underwear.

It is very frustrating. We have tried bribes, reason, trying to get him to help clean it, etc. but nothing has worked so far. We have books upon books. We talk about pooping and invite him when we poop. We have tried pooping games and talking silly about it. Talked about kids at daycare pooping.

He is insistent he doesn’t want to, even when we bring it up when he is not upset. He says he will when he is four. He says he is scared of pooping and also that he doesn’t know when he needs to. I do see him get scared when he’s sitting and he feels it coming. But then he is scared and will of course not relax to sit on the potty.

Looking for any advice. We are probably going to try the pants free method in a week or two if it keeps going like this. My husband and I just want a solid plan if it’s the case. We figured we will need to tackle nighttime at the same time if this is the case because we don’t want to have him hold it until diapers at bedtime.