Generic torsion vanishing conjecture for Shimura varieties

Let $(\mathbf{G},X)$ be a Shimura datum such that $G=\mathbf{G}_{\mathbb{Q}_{p}}$ is unramified, and let $K=K_{p}K^{p}$ be a sufficiently small level with $K_{p}=K_{p}^{\mathrm{hs}}$ hyperspecial. Set $d=\operatorname{dim}(\mathrm{Sh}(\mathbf{G},X)_{K})$. A maximal ideal $\mathfrak{m}\subset H_{K_{p}^{\mathrm{hs}}}$ is generic when its associated toral $L$-parameter satisfies the genericity condition described in the surrounding text. Generic torsion vanishing conjecture. If $\mathfrak{m}$ is generic, then the cohomology of

$$R\Gamma(\mathrm{Sh}(\mathbf{G},X)_{K,\overline{E}},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}}$$

is concentrated in degrees $d\leq i\leq 2d$, while the cohomology of

$$R\Gamma_{c}(\mathrm{Sh}(\mathbf{G},X)_{K,\overline{E}},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}}$$

is concentrated in degrees $0\leq i\leq d$. This predicts torsion-vanishing ranges for localized cohomology under a genericity hypothesis; the source motivates it by known results for general linear groups, but supplies no resolution.