Root compatibility conjecture for abelian mirror branes

Let $Y$ be a Hamiltonian $G$-manifold with maximal torus $T$, Weyl group $W$, and root system. Let $(Y^\vee,f)$ be its two-dimensional mirror Landau–Ginzburg model, let $\tau_T:Y^\vee\to\check T_\mathbb{C}$ be the associated holomorphic map, and let

$$Z^T_Y=\{(p,z,a)\in Y^\vee\times\check T_\mathbb{C}\times\mathfrak{t}_\mathbb{C}:z=\tau_T(p),\ df|_p=\tau_T^*da|_p\}.$$

For a root $\alpha$, write $\alpha^\vee$ for its coroot. Root compatibility conjecture. For every root $\alpha$ and every $(p,z,a)\in Z^T_Y$ satisfying $\alpha(a)=0$, one has

$$\alpha^\vee(\tau_T(p))=\alpha^\vee(z)=1.$$

This condition is stated as equivalent to the inclusion of the Weyl-group Hilbert-scheme construction in the Coulomb branch. The source does not provide a proof or disproof.