Cut-at-vertex Bernardi conjecture for the exterior polynomial

Let $\mathcal H$ be a connected hypergraph with a ribbon structure, base node, and base edge. Let $\tilde{X}_{\mathcal H}$ be the exterior-inactivity polynomial defined from the (ht:$E$, cut:$E$) Bernardi process, and let $X_{\mathcal H}$ be the exterior polynomial. One may alternatively define $\tilde{X}_{\mathcal H}$ using the (ht:$E$, cut:$V$) Bernardi process.

Cut-at-vertex exterior-polynomial conjecture. For every such $\mathcal H$, $\tilde{X}_{\mathcal H}=X_{\mathcal H}$, $\tilde{X}_{\mathcal H}$ is independent of the ribbon structure, and both assertions remain true after redefining $\tilde{X}_{\mathcal H}$ using the (ht:$E$, cut:$V$) process.

This is the exterior-polynomial counterpart of the preceding interior-polynomial results and is stated without a resolution in the source.