The weak local Langlands interpolation conjecture for framed deformation rings

Let $L$ be the local field, let $\overline{\rho}$ be a residual representation, let $R_{ \overline{\rho}}^{\Box}$ be its framed deformation ring, and let $(A_{[L,\pi]})_{\mathfrak m}$ be the relevant localization of the integral Bernstein center. A map from $(A_{[L,\pi]})_{\mathfrak m}$ to $R_{ \overline{\rho}}^{\Box}$ is an interpolating map. It interpolates the characteristic-zero semisimple local Langlands correspondence when, on characteristic-zero points, it sends a framed deformation $\rho$ to the point corresponding to its semisimple local Langlands parameter.

Weak local Langlands interpolation conjecture. For any $\overline{\rho}$, there is a map

$$\operatorname{LL}: (A_{[L,\pi]})_{\mathfrak m} \rightarrow R_{ \overline{\rho}}^{\Box}$$

that interpolates the characteristic-zero semisimple local Langlands correspondence.

This conjecture is motivated by the preceding interpolation proposition and, conditional on the stated proposition, follows from the existence of a universal representation $\pi(\rho^{\Box})$. The source gives no resolution status.