Armor does not render properly on Zombie villagers

Summary:
The clothing of zombie villagers renders through armor. The helmet is positioned too low (the head goes through the helmet). This is almost the same problem as MC-91283 (there is no problem with helmet rendering in Java).

Steps to reproduce:

1. Put a full set of armor into a dispenser.
2. Point that dispenser into a Zombie Villager, power the dispenser.

Observed results:
The zombie villager's texture is visible through its armor.

Expected results:
Zombie villager's texture will not be rendered over armor.

Screenshots: