I found a bug that can reset conversion cooldown (I tried with Japanese, not sure if it works with other civs). Need 2 monks and 1 monastery to do this:
Step 1: 1 monk to pick up the relic and use conversion as normal then drop that relic in the monastery.
Step 2: Immediately pick it up with the same monk then drop it on the ground.
Step 3: Use the other monk to pick up the relic, it will reset the conversion cooldown.
I found this very strong when defending against melee comp, hope it gets fix soon.