Temperedness conjecture for global packets on basic inner forms

Let $G^*$ be a connected, reductive quasi-split group, let $(G,\psi)$ be an inner form of $G^*$, and let

$$\mathscr{T}\in Z^1(\textnormal{Kott}_F^{1/\infty},G)_{\mathrm{basic}}$$

be an enrichment, with associated family of local $B_e(G^*_{F_v})_{\mathrm{basic}}$-enhanced inner forms $\{(G_{F_v},\psi_v,\mathscr{T}_v,h_v)\}_{v\in V_F}$. Define $\Pi_\phi$ as in the source's global-packet construction.

Temperedness conjecture. The set $\Pi_\phi$ always consists of irreducible, admissible, and tempered representations of $G(\mathbb A_F)$.

The conjecture strengthens the preceding lemma, which already asserts these properties for the globally defined packet in the setting considered there. It is not needed for the paper's main purposes, and the source gives no resolution status.