Cured zombie villagers despawn and will not breed

What I have been trying to do is cure zombie villagers that come from a zombie spawner, breed them, and bring them to a deserted village. I am able to separate the zombie villagers, cure them, and move them into a large room full of doors for breeding. I cured four and moved them to the room. But, no matter how long I leave them there, they do not breed. Additionally, whenever I leave the area, they despawn. It seems like, even though they have been turned back into villagers, they retain the nonbreeding and despawning traits of a zombie.