Coherence conjecture for the free cornering with choice and iteration

Let $\mathbb{A}$ be a distributive monoidal category. Write \mathbf{V}\,{}^\ulcorner_\llcorner\!{\mathbb{A}}\!_\lrcorner^\urcorner^\oplus and \mathbf{V}\,{}^\ulcorner_\llcorner\!{\mathbb{A}}\!_\lrcorner^\urcorner^* for the categories of vertical cells in the free cornering with choice and the free cornering with iteration, respectively. Coherence conjecture. There are isomorphisms of categories

\mathbf{V}\,{}^\ulcorner_\llcorner\!{\mathbb{A}}\!_\lrcorner^\urcorner^\oplus \cong \mathbb{A}

and

\mathbf{V}\,{}^\ulcorner_\llcorner\!{\mathbb{A}}\!_\lrcorner^\urcorner^* \cong \mathbb{A}.

This conjecture extends the known coherence result for the original free cornering, whose vertical-cell category is isomorphic to the base category. The source provides no resolution, so both asserted isomorphisms remain open.