The alternating and c-noncrossing face conjecture for prepolypositroids

Let $W$ be a Weyl group, let $c$ be a Coxeter element, and let $P$ be a generic simple $(W,c)$-prepolypositroid. Let $\widetilde R$ be the associated set of transformed roots, and let $(\widetilde\beta,\widetilde\gamma)\in\widetilde R$ span a two-dimensional face of the normal fan of $P$. Two roots $(\beta,\gamma)$ are alternating when $(\beta,\gamma)=(\gamma,\beta)\geq 0$, and they are $c$-noncrossing when either $(\gamma^\vee,\widetilde\beta)=0$ or $(\beta^\vee,\widetilde\gamma)=0$.

Alternating and c-noncrossing face conjecture. If $(\widetilde\beta,\widetilde\gamma)\in\widetilde R$ span a two-dimensional face of the normal fan of $P$, then $(\beta,\gamma)$ must be alternating and $c$-noncrossing.

This conjecture predicts a necessary combinatorial condition on pairs of roots associated with two-dimensional faces of normal fans of generic simple $(W,c)$-prepolypositroids. The supplied text gives no resolution status.