Drinfeld's conjecture on smooth sheaves on simply connected BunG

Let $G^{sc}$ be the simply connected cover of the derived group of a reductive group $G$, and let $\operatorname{Bun}_{G^{sc}}$ be the stack of $G^{sc}$-torsors on $X$. A smooth $\mathbb{\bar Q}_\ell$-sheaf on this stack is a lisse $\ell$-adic sheaf. Drinfeld's conjecture. Any smooth $\mathbb{\bar Q}_\ell$-sheaf on $\operatorname{Bun}_{G^{sc}}$ is constant. The conjecture would imply that the almost constant local systems arising in the study of the minimal representations have no nontrivial variation after pullback to $\operatorname{Bun}_{G^{sc}}$. The source reports no proof and notes that an $\ell$-adic version of results of Gaitsgory would imply it.