Calegari–Emerton vanishing conjecture for completed cohomology

Let $G$ be a reductive group over $\mathbb{Q}$ admitting a Shimura datum $(G,X)$, and let $(\operatorname{Sh}_{K,\mathbb{C}})_K$ be the associated inverse system of Shimura varieties. Fix a compact open subgroup $K^p\subseteq G(\mathbb{A}_f^p)$ and write $d$ for the common dimension of $\operatorname{Sh}_{K,\mathbb{C}}$. Emerton's completed cohomology is denoted by $\widetilde{H}^q(K^p,\mathbb{Z}_p)$. Calegari–Emerton vanishing conjecture. For every integer $q>d$, one has

$$\widetilde{H}^q(K^p,\mathbb{Z}_p)=0.$$

This conjecture predicts that completed cohomology is concentrated in degrees at most the dimension of the Shimura varieties, as expected from its role in the $p$-adic Langlands program. Its resolution is not specified in the source.