The Satake compatibility conjecture for the spectral action

Let $\boldsymbol{J}$ be the algebra acting on the compactly supported cohomology of the Shimura variety, and let \operatorname{H}_c(\operatorname{Sh}_{\mu,\overline{\mathbb{F}}_v,\mathbb{Q}_\ell) denote that cohomology. The classical Satake isomorphism identifies $\boldsymbol{J}$ with the relevant spherical Hecke algebra. Satake compatibility conjecture. Under the Satake isomorphism, the action of $\boldsymbol{J}$ in the preceding theorem coincides with the usual Hecke algebra action on

$$\operatorname{H}_c^*(\operatorname{Sh}_{\mu,\overline{\mathbb{F}}_v},\mathbb{Q}_\ell).$$

This is presented as the analogue of V. Lafforgue's $S=T$ theorem and would identify the geometric spectral action with the classical Hecke action. The source gives no resolution.