Galois equivariance of the semisimple local Langlands correspondence

Suppose the semisimple local Langlands correspondence $bbLL_{bbG}$ exists. Let $bbsigma$ be an automorphism of $bbC$ fixing $bqsqrt q$, and let $(bbM,bbrho)$ be a supercuspidal-support datum for $bbG$. Galois-equivariance conjecture. One has

$$bbLL_{bbG}(bbrho^{bbsigma})=bbLL_{bbG}(bbrho)^{bbsigma}.$$

The source introduces this conjecture to descend results to the smallest possible base ring. Its status is not resolved in the supplied text.