Vertex-preservation conjecture for the type D string-to-Gelfand–Tsetlin map

Let $\lambda$ be a dominant weight for a complex classical group of type $\mathsf{D}_n$. Let $\mathcal{Q}_{\underline{w_0}^\mathrm{std}}(\lambda)$ be the standard string polytope, let $GT_{\mathsf{D}_n}(\lambda)$ be the type $\mathsf{D}_n$ Gelfand–Tsetlin polytope, and let $\phi_\lambda$ be the piecewise affine bijection satisfying

$$\phi_\lambda\bigl(\mathcal{Q}_{\underline{w_0}^\mathrm{std}}(\lambda)\bigr)=GT_{\mathsf{D}_n}(\lambda).$$

For a polytope $P$, write $\operatorname{vert}P$ for its set of vertices.

Vertex-preservation conjecture. The map $\phi_\lambda$ induces a bijection

$$\operatorname{vert}\mathcal{Q}_{\underline{w_0}^\mathrm{std}}(\lambda)\longrightarrow\operatorname{vert}GT_{\mathsf{D}_n}(\lambda).$$

If true, this would address the obstruction in type $\mathsf{D}_n$ caused by the fact that the known piecewise affine bijection need not a priori preserve vertices, and would help establish the corresponding string-polytope results.