Langlands-Shahidi type categorical Langlands conjecture

Fix a quasi-split reductive group $G$ over a local field $E$, and let $W_E$ be its Weil group. Let $\check{G}$ be the Langlands dual group, and let $Z^1_{\mathrm{LSt}}(W_E,\check{G})/\check{G}$ and $Z^1_{\mathrm{wLSt}}(W_E,\check{G})/\check{G}$ denote the stacks of Langlands-Shahidi type and weakly Langlands-Shahidi type parameters. Fix a Whittaker datum $(U,\psi)$ and let $\mathcal{W}\coloneqq i_{1!}\operatorname{c-ind}_{U(E)}^{G(E)}\psi$. Langlands-Shahidi type categorical Langlands conjecture. There is an equivalence of categories

$$\operatorname*{Ind}\operatorname{Perf}^{\mathrm{qc}}(Z^1_{\mathrm{LSt}}(W_E,\check{G})/\check{G})\simeq\operatorname{\mathcal{D}}^{\mathrm{LSt}}(\operatorname{Bun}_G)$$

that maps the structure sheaf $\mathcal{O}$ to $\mathcal{W}^{\mathrm{wLSt}}$, is linear for the spectral action of $\operatorname{Perf}(Z^1_{\mathrm{wLSt}}(W_E,\check{G})/\check{G})$ on both sides, and is $t$-exact for the good filtration $t$-structure on the left and the perverse $t$-structure on the right. The same assertion holds after replacing $\mathrm{wLSt}$ everywhere with $\mathrm{LSt}$. This is a refinement of the $d$-adic categorical Langlands conjecture to Langlands-Shahidi type parameters; the stated $t$-exactness is related to right $t$-exactness of Hecke operators for representations with good filtration. The conjecture remains open.