Weak Langlands–Shahidi torsion vanishing conjecture for Shimura varieties

Let $\phi$ be a semi-simple $L$-parameter of weakly Langlands–Shahidi type with cuspidal support $(M,\phi_{M})$. Let $\mathcal{S}(\mathbf{G},X)_{K^{p},C}$ be the corresponding Shimura variety space at tame level $K^{p}$ and compact-open coefficient datum $C$, let $\Lambda$ be the coefficient ring, and let $d$ denote the Shimura variety dimension. Weak Langlands–Shahidi torsion vanishing conjecture. The compactly supported complex

$$R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\Lambda)_{\phi}$$

is concentrated in degrees $0\leq i\leq d$, whereas

$$R\Gamma(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\Lambda)_{\phi}$$

is concentrated in degrees $d\leq i\leq 2d$. This is presented as a consequence of the preceding geometric conjecture together with a generalization of another theorem to arbitrary Shimura varieties; the source gives no resolution.