Equivariant subobject-classifier conjecture for the contextuality topos

Let $cmathcal{V}(cmathcal{H})$ be the poset of contexts of the Hilbert space $cmathcal{H}$, let $cmathcal{V}_f(cmathcal{H})$ denote the corresponding context category with the group action forgotten, and let $F$ be the functor relating the associated sheaf topoi. Write $\underline{\Omega}^{cmathcal{V}(cmathcal{H})}$ for the subobject classifier and $\underline{G/G_F}$ for the corresponding constant equivariant object. Equivariant subobject-classifier conjecture. There is an isomorphism

$$F(\underline{\Omega}^{\mathcal{V}(\mathcal{H})})\simeq\underline{G/G_F}\times\underline{\Omega}^{\mathcal{V}(\mathcal{H})}.$$

The claim identifies the image of the subobject classifier under $F$ with the product of the orbit object and the original subobject classifier; the source immediately supplies a proof, so this candidate is solved rather than open.