Property taxes of rental house which provides housing allowance

by LCM
(Chicago, IL USA)

I live in Cook County, Illinois. Our church currently has a pastoral house which provides as the residence for our pastor.

However, the pastor desires to move to a different neighborhood with his family, in which case we would rent out the pastoral house and provide him with that money as a housing allowance.

Would the church have to now pay property taxes on the pastoral house that is being rented out?

Arguably, the pastoral house is still technically being used for the benefit and purpose of the priest.

Thank you in advance for your time in answering this question.