Toric homological mirror symmetry monodromy conjecture

Let $\check X$ be a toric variety with $d$ rays in its fan, and let $T$ be an algebraic torus of rank $\dim(\check X)$. Let $V(E_A)$ be the hypersurface cut out by the principal $A$-determinant of a matrix $A$ determined by the toric GIT problem. Toric HMS monodromy conjecture. There is an action

$$\pi_1\left(\frac{\mathbb A^d\setminus V(E_A)}{T}\right)\longrightarrow \operatorname{Auteq}\left(D^b(\check X)\right).$$

This proposes the categorical monodromy action associated with the toric mirror-moduli space, but the source does not state a resolution of the conjecture.