Identification of the sheaf-theoretic and KZ functors

Let $ccalC_0$ and $ccalB_0$ be the source and target categories, let $ccalM:ccalC_0\longrightarrow ccalB_0$ be the sheaf-theoretic functor, and let $cPhi_*$, $cKZ$, and $cPsi^*$ be the functors appearing in the paper. Functor-identification conjecture. The functors

$$\calM:\calC_0\longrightarrow \calB_0$$

and

$$\Psi^*\circ \KZ\circ \Phi_*:\calC_0\longrightarrow \calB_0$$

are isomorphic.

This assertion identifies the sheaf-theoretic functor with the composite obtained from the geometric equivalence and the KZ functor. The supplied context does not state whether the assertion is proved, so its status is left open.