The compact induction conjecture for irreducible supercuspidal representations

Let $G$ be a reductive group and let $\mathcal{M}(G)$ denote the representations of $G$. Let $\pi\in\mathcal{M}(G)$ be irreducible and supercuspidal. A subgroup $K$ of $G$ is compact modulo the centre if its image in the quotient of $G$ by the centre is compact. Let $\mathrm{c\text{-}ind}_K^G(\rho)$ denote compact induction from $K$ to $G$. Compact induction conjecture. There exist an open subgroup $K$ compact modulo the centre and an irreducible smooth representation $\rho$ of $K$ such that

$$\pi=\mathrm{c\text{-}ind}_K^G(\rho).$$

This conjecture asserts that all irreducible supercuspidal representations of reductive groups arise by the compact-induction construction described in the surrounding discussion. The source does not state a resolution, so the conjecture is recorded as open.