The compact induction conjecture for supercuspidal representations

Let $F$ be a nonarchimedean local field, let $\boldsymbol{\mathrm{G}}$ be a connected reductive group defined over $F$, and let $G=\boldsymbol{\mathrm{G}}(F)$ with center $Z$. Let $(\pi,V)$ be an irreducible supercuspidal representation of $G$. The compact induction conjecture. There exists a compact subgroup $K\subset G$ and a $K$-stable subspace $W\subset V$ such that

$$(\pi,V)=\operatorname{cInd}_{KZ}^G W$$

as representations of $G$, where $W$ is viewed as a representation of $KZ$ by letting $Z$ act through the central character of $(\pi,V)$. This is a folklore conjecture in the representation theory of $p$-adic groups; the paper proves it for groups of relative rank one, while the general case remains open.