Galois invariance of cuspidal multiplicities

Let $G$ be the reductive group in the source, let $K$ be the chosen model of a maximal compact subgroup, and let $\Pi$ be an irreducible cohomological cuspidal automorphic representation of $G(\mathbb{A})$. For $\sigma\in\operatorname{Aut}(\mathbb{C}/\mathbb{Q}_K)$, let $\Pi_\infty^\sigma$ and $\Pi_{(\infty)}^\sigma$ denote the twisted archimedean and finite components, and let $m_{L^2_0}$ denote the multiplicity in the cuspidal $L^2$-spectrum. Galois multiplicity conjecture. For every such $\sigma$,

$$m_{L^2_0}(\Pi_\infty^{\rm CW}\otimes\Pi_{(\infty)})=m_{L^2_0}(\Pi_\infty^\sigma\otimes\Pi_{(\infty)}^\sigma).$$

This is presented as a stronger conjecture that would imply equality of multiplicities under twisting; the source does not give a resolution and treats it as open.