Curtis's strong conjecture on the Bernstein center and invariant deformation rings

Let $F$ be a non-archimedean local field, let $G_n=\operatorname{GL}_n(F)$, let $Z_n$ be the Bernstein center, and let $e$ be the primitive idempotent associated with an inertial type $\nu$. Let $R_\nu$ be the universal finite-type algebra for the corresponding suitably rigidified Weil-group representations, equipped with an action of an algebraic group $G_\nu$, and write $R_\nu^{\operatorname{inv}}$ for its invariant subring. Let $\mathbb L_\nu:eZ_n\to R_\nu$ be the map from the weak conjecture. Curtis's strong conjecture. The map $\mathbb L_\nu$ identifies $eZ_n$ with $R_\nu^{\operatorname{inv}}$. This strengthens the weak conjecture by asserting that the Bernstein-center block is exactly the invariant coordinate ring of the universal deformation space; the assertion is open in the source.