Strong local Langlands correspondence in families

With the notation above, let $R_\nu^{\operatorname{inv}}$ denote the subalgebra of $\mathcal G_\nu$-invariant elements of $R_\nu$. Strong local Langlands in families. There is an isomorphism

$$Z_{[L,\pi]}\cong R_\nu^{\operatorname{inv}}$$

such that the composition

$$Z_{[L,\pi]}\longrightarrow R_\nu^{\operatorname{inv}}\longrightarrow R_\nu$$

is compatible with local Langlands. This conjecture describes the relevant Bernstein-center factor purely in terms of Galois deformation data; the supplied material gives no resolution status.