We went simultaneously. Both of us to separate IC and then to a joint MC. I consider our MC to have been our marriages savior. He helped us tremendiously. That being said, I had a FWH that wanted to do everything in his power to R. Perhaps your WH is afraid that a MC will, righteously, hold him accountable for his A.
I would be inclined to give a bit and wait another month or so for the MC. But if he hasn't given the full truth, then I would not budge on the lie detector test.