The strong local Langlands interpolation conjecture for semisimple residual representations

Let $\overline{\rho}$ be a semisimple residual representation, let $R_{ \overline{\rho}}^{\Box}$ be its framed deformation ring, let $G^{\Box}$ be the formal completion of $\operatorname{GL}_n/W(k)$ at the identity, and let $(A_{[L,\pi]})_{\mathfrak m}$ be the relevant localized integral Bernstein center. The group $G^{\Box}$ acts on $R_{ \overline{\rho}}^{\Box}$ by changing the frame, and $(R_{ \overline{\rho}}^{\Box})^{G^{\Box}}$ denotes the invariant subring.

Strong local Langlands interpolation conjecture. If $\overline{\rho}$ is semisimple, there is a unique isomorphism

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

that interpolates the characteristic-zero semisimple local Langlands correspondence.

This is presented as a stronger version of the preceding weak conjecture, using the naturality of the integral Bernstein-center and invariant deformation-ring constructions. The source gives no resolution status.