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.