The support and restriction conjecture for tame categorical local Langlands sheaves

Let $Z^1(W_F,\widehat{G})$ be the moduli of L-parameters over $\overline{\mathbb{Q}}_{\ell}$ for $\mathbf{G}$, let $C_{[\varphi]}$ be the connected components containing a lift of $[\varphi]$, and let

$$u_G \colon Z^1(W_F,\widehat{G}) \to \mathcal{N}_{\widehat{G}}$$

be the unipotent monodromy morphism, where $\mathcal{N}_{\widehat{G}}$ is the nilpotent cone of $\operatorname{Lie}(\widehat{G})$. Let $N_{[\varphi]}$ be the image of the set of lifts of $[\varphi]$, write $\overline{N}_{[\varphi]}$ for its closure in $\mathcal{N}_{\widehat{G}}$, and define

$$X_{\leq [\varphi]}=C_{[\varphi]}\cap u_G^{-1}(\overline{N}_{[\varphi]}).$$

Let $\mathcal{S}_{\pi}$ be the coherent sheaf on the moduli of L-parameters given by $\operatorname{c-Ind}_{\mathcal{G}(\mathcal{O}_F)}^{\mathbf{G}(F)}\pi$ under the tame categorical local Langlands correspondence. Let $X_{[\varphi]}$ be the image of the torsor $h_{\widetilde{\varphi}}$ defined from a lift $\widetilde{\varphi}$ of $[\varphi]$, and let $V(\widetilde{\rho}_{\pi})$ be the associated vector bundle determined by the inflated representation $\widetilde{\rho}_{\pi}$. The support and restriction conjecture. The support of $\mathcal{S}_{\pi}$ is contained in $X_{\leq [\varphi]}$, and the restriction of $\mathcal{S}_{\pi}$ to $X_{[\varphi]}$ is isomorphic to $V(\widetilde{\rho}_{\pi})$.

This predicts both a geometric support bound and an explicit description on the distinguished stratum in the tame categorical local Langlands correspondence. The parser provides no evidence that the statement has been resolved.