The mixed tensor product conjecture for Verma modules of $\mathfrak{gl}(m+1|n)$

Let $\xi\in\mathfrak{h}_{m+1|n}^*$ be any weight, and let

$$\mathscr{F}(\xi)=\mathcal{F}_{\xi_0}\otimes\mathcal{F}_{\xi_1}\otimes\cdots\otimes\mathcal{F}_{\xi_{\bar n}}$$

be the $\mathfrak{gl}(m+1|n)$-module constructed from the modules $\mathcal{F}_a$. Here $M(\xi)$ denotes the Verma module of highest weight $\xi$.

Mixed tensor product conjecture. There is an isomorphism of $U(\mathfrak{gl}(m+1|n))$-modules

$$\mathscr{F}(\xi)\cong M(\xi)$$

for all weights $\xi\in\mathfrak{h}_{m+1|n}^*$.

For antidominant typical weights, the preceding construction identifies $\mathscr{F}(\xi)$ with both the simple module $L(\xi)$ and the Verma module $M(\xi)$. The conjecture asks whether this realization extends to arbitrary weights, substantially beyond the antidominant typical setting.