Zhu's unramified categorical local Langlands conjecture

Let $u$ be a closed point of $U$, let $\LS_{\prescript{L}{}{G},F_u}$ be the local Langlands parameter stack, and let $\LS_{\prescript{L}{}{G},\cO_u}$ denote its unramified locus. Let $\bL_{\psi_u}$ be the Fargues–Scholze functor, and let $\cInd_{G(\cO_u)}^{G(F_u)}\La$ be the unramified compact induction. Zhu's conjecture. Assume Fargues–Scholze's conjecture for $u$. Then

$$\bL_{\psi_u}(i_{1,!}\!\cInd_{G(\cO_u)}^{G(F_u)}\La)\cong\sO_{\LS_{\prescript{L}{}{G},\cO_u}}.$$

Moreover, there is a natural isomorphism of $\bE_1$-algebras over $\La$,

$$\End_{G(F_u)}(\cInd_{G(\cO_u)}^{G(F_u)}\La)\cong\End_{D_{\coh}(\LS_{\prescript{L}{}{G},F_u})}(\sO_{\LS_{\prescript{L}{}{G},\cO_u}}).$$

The first assertion implies the second. The first is known when $G$ is a torus, while the second is known when $\La$ equals $L$; the conjecture in general remains open.