Categorical Langlands conjecture for the full derived category

Let $G$ be a quasi-split reductive group over $E$, let $\check{G}$ be its Langlands dual group, and assume $\ell\nmid\lvert\pi_0(Z(G))\rvert$. Let $W_E$ be the Weil group and let $Z^1(W_E,\check{G})/\check{G}$ be the stack of parameters. Fix a Whittaker datum $(U,\psi)$ and let $\mathcal{W}\coloneqq i_{1!}\operatorname{c-ind}_{U(E)}^{G(E)}\psi$. Categorical Langlands conjecture. For the functor

$$-*\mathcal{W}\colon\operatorname*{Ind}\operatorname{Perf}^{\mathrm{qc}}(Z^1(W_E,\check{G})/\check{G})\to\operatorname{\mathcal{D}}(\operatorname{Bun}_G)$$

given by the spectral action on $\mathcal{W}$, the right adjoint $c_{\mathcal{W}}$ is fully faithful on $\operatorname{\mathcal{D}}(\operatorname{Bun}_G)^\omega$, and the essential image of $\operatorname{\mathcal{D}}(\operatorname{Bun}_G)^\omega$ under $c_{\mathcal{W}}$ is $\operatorname{Coh}_{\mathrm{Nilp}}(Z^1(W_E,\check{G})/\check{G})$. This is the full $d$-adic categorical Langlands statement recalled before the paper's specialization to Langlands-Shahidi type parameters. Its resolution is not indicated in the supplied text, so it remains open.