My state refund says I owe even though state taxes were withheld correctly, but local taxes were not withheld by my employer correctly. Will that total go to local?

To elaborate, due to a switch in tax companies my employer made early in the year, they failed to withhold my taxes at the local level for 11 months. As a result, my state refund says I will owe an amount close to what I would owe at the local level, however my state taxes were withheld correctly so I assume the amount is for my local taxes, and that if I pay that money it will go to the local level correctly.

Is this safe to assume?