Cuspidality conjecture for the categorical local Langlands correspondence

Let $G$ be a reductive group, let $B(G)_{\operatorname{bas}}$ denote the basic $\sigma$-conjugacy classes, and let $G_b$ be the extended pure inner form corresponding to $b\in B(G)_{\operatorname{bas}}$. Write $i^b\colon \operatorname{Bun}_G^b=[\ast/G_b(E)]\hookrightarrow \operatorname{Bun}_G$ for the open immersion, and let $\operatorname{CT}_P$ be the normalized geometric constant term functor for a parabolic subgroup $P$ of $G$. Define $\mathcal{D}_{\operatorname{lis}}(\operatorname{Bun}_G,\overline{\mathbb{Q}}_\ell)_{\operatorname{cusp}}$ as the subcategory of objects $A$ such that $\operatorname{CT}_P(A)=0$ for every proper parabolic subgroup $P$ of $G$. Cuspidality conjecture. The inclusion

$$\mathcal{D}_{\operatorname{lis}}(\operatorname{Bun}_G,\overline{\mathbb{Q}}_\ell)_{[G]}\subset \mathcal{D}_{\operatorname{lis}}(\operatorname{Bun}_G,\overline{\mathbb{Q}}_\ell)_{\operatorname{cusp}}$$

is an equality. In particular, for every smooth irreducible $\overline{\mathbb{Q}}_\ell$-representation $\pi$ of $G_b(E)$, its Fargues--Scholze parameter $\varphi^{\operatorname{FS}}_\pi$ is supercuspidal if and only if

$$\operatorname{CT}_P(i^b_!\pi)=0$$

for every proper parabolic subgroup $P$ of $G$. This conjecture identifies the categorical $[G]$-part with the cuspidal subcategory and predicts a corresponding characterization of supercuspidal Fargues--Scholze parameters. The inclusion is established in the source; the converse, and hence the equality, is conjectural.