Hecke equivariance conjecture for base change on cohomology

Let $c$ be a $\mathbf{G}$-Hecke eigenclass with eigenvalues $a_\rho$, and let $\widetilde{C}$ be the graded $\mathbb{Z}[\sigma]$-submodule predicted by the Galois structure conjecture. For every place $v$ where $\mathbf{G}/F_v$ is split and $J_v$ is hyperspecial, write

$$T_\rho(c)=a_\rho c$$

for all $T_\rho\in\mathcal{H}_v\cong\operatorname{Rep}({}^L\mathbf{G})$. Let $\phi$ denote the base-change map on $L$-groups, and let $\widetilde{c}\in\widetilde{C}$.

Hecke equivariance conjecture. One has

$$T_{\widetilde{\rho}}\widetilde{c}=a_{\widetilde{\rho}\circ\phi}\widetilde{c}.$$

This is presented as an even more speculative conjecture concerning Hecke equivariance of the cohomological correspondence. The source gives no resolution.