The conjecture on the Satake action of the algebra $\mathbf J$

Let $\Sh_{\mu,\overline{\mathbb F}_v}$ be the special fiber of the relevant Shimura variety, let $\Ql$ denote the coefficient field, and let $\mathbf J\subset {\operatorname{End}}(\widetilde{V_{\mu}})$ be the algebra acting on its compactly supported cohomology. Under the Satake isomorphism, compare this action with the usual Hecke algebra action on

$${\operatorname{H}}_c^{*}(\Sh_{\mu,\overline{\mathbb F}_v},\Ql).$$

The Satake-action conjecture. Under the Satake isomorphism, the action of $\mathbf J$ coincides with the usual Hecke algebra action on

$${\operatorname{H}}_c^{*}(\Sh_{\mu,\overline{\mathbb F}_v},\Ql).$$

The paper presents this as the analogue of V. Lafforgue's $S=T$ theorem, following a preceding theorem describing the action of $\mathbf J$; the source gives no resolution status.