The Hecke compatibility conjecture for the spectral action on Shimura cohomology

Let $\boldsymbol{J}$ be the algebra of regular functions on the stack $[\hat G\sigma_p/\hat G]$, acting on $\operatorname{H}_c(\operatorname{Sh}_{\mu,\overline{k}_v})$, and identify it with the spherical Hecke algebra through the Satake isomorphism. Hecke compatibility conjecture. The action of $\boldsymbol{J}$ on $\operatorname{H}_c(\operatorname{Sh}_{\mu,\overline{k}_v})$ constructed above coincides with the usual Hecke algebra action, via the Satake isomorphism.

This conjecture asserts compatibility between the newly constructed spectral action and the classical spherical Hecke action on Shimura-variety cohomology. It is a restatement of the preceding Satake-compatibility claim in the later notation, so the two candidate spans represent one conjecture.