Principal A-determinant stratification for the triple Segre product

Let $w$ and $w'$ be scalings for the Segre embedding of $\mathbb P^1\times\mathbb P^1\times\mathbb P^{n}$, and let $P_n$ denote the relevant polytope. For every face $\Gamma\subseteq P_n$, let $\Delta_\Gamma(w)$ be the corresponding principal $A$-determinant factor. Principal A-determinant stratification conjecture. If

$$\Delta_\Gamma(w)=0\quad\Longleftrightarrow\quad\Delta_\Gamma(w')=0$$

for every face $\Gamma\subseteq P_n$, then

$$\operatorname{mldeg}(X_w)=\operatorname{mldeg}(X_{w'}).$$

For the two-factor Segre embedding, the ML degree is already determined by the vanishing factors of the principal $A$-determinant; the conjecture asserts the analogous statement for the triple product, which remains open.