The center conjecture for Whittaker coinvariants of $\mathrm{GL}(m|n)$

Let $W$ be the Weyl group and let $C_\xi$ be the center of the block $\mathcal O_\xi$, identified with the center of the completion of its Soergel algebra. Let $B_\xi\cap C_\xi$ denote the intersection inside the completed algebra, and let $Z(W)\rightarrow C_\xi$ be the canonical map.

Center conjecture. The image of the canonical map

$$Z(W)\longrightarrow C_\xi$$

is isomorphic to

$$C_\xi/(B_\xi\cap C_\xi).$$

This predicts a description of the part of the center coming from the Weyl-group center; the supplied text gives no resolution.