Suppose $\mathcal{C}$ is a unitary ribbon fusion category. Also assume that its symmetric centre has trivial twist and trivial pivotal structure, i.e. is tannakian. Thus, the Müger/Bruguières modularisation/deequivariantisation exists.

*Is the resulting modular fusion category unitary? Is the modularisation functor unitary (dagger)?*

It feels to me like it should be obviously true, coming from the philosophy that the modularisation is a kind of generalised fibre functor. But I can't find a reference, nor can I write down the dagger structure.