The conjecture on embedding-independent temperedness

Let $G$ be a reductive group over the global function field $K$, let $[\Pi]$ be a cuspidal automorphic representation of $G([\mathbf A_K])$ with coefficients in $\bar{\mathbb Q}$, and let $[\iota: \bar{\mathbb Q}\hookrightarrow\mathbb C]$ be an embedding. The representation $[\Pi_v]$ is $[\iota]$-tempered if $[\Pi_v\otimes_{\bar{\mathbb Q},\iota}\mathbb C]$ is tempered, and it is tempered if it is $[\iota']$-tempered for every embedding $[\iota']$. Embedding-independent temperedness conjecture. If $[\Pi_v]$ is $[\iota]$-tempered, then $[\Pi_v]$ is tempered. The source says this should follow from the Arthur conjectures but that the authors have not proved it; its general resolution status is therefore open.