Curtis's weak conjecture on the Bernstein center and finite-type deformation algebras

Let $F$ be a non-archimedean local field, let $G_n=\operatorname{GL}_n(F)$, let $Z_n$ be the Bernstein center of the category of smooth representations of $G_n$, and let $\nu$ be a representation of the prime-to-$\ell$ inertia group $I_F^{(\ell)}$. Let $e$ be the primitive idempotent of $Z_n$ corresponding to the inertial type $\nu$. Let $R_\nu$ be the finite-type $W(k)$-algebra parametrizing the relevant suitably rigidified representations of the Weil group, with universal representation $\rho_\nu:W_F\to\operatorname{GL}_n(R_\nu)$. For a point $x:R_\nu\to\overline{\mathcal K}$, write $\rho_{\nu,x}$ for the specialization and let $\Pi_x$ be the representation corresponding to it under local Langlands. Curtis's weak conjecture. There is a map

$$\mathbb L_\nu:eZ_n\longrightarrow R_\nu$$

compatible with local Langlands, in the sense that for every $x:R_\nu\to\overline{\mathcal K}$, the composition with $\mathbb L_\nu$ is the map $eZ_n\to\overline{\mathcal K}$ giving the action of $eZ_n$ on $\Pi_x$. This conjecture is a finite-type refinement of the local Langlands correspondence in families and is formulated in terms of the Bernstein center; the map is unique if it exists, but its existence is not established here.