Conjecture on conjugacy of cuspidal types

Let $G$ be an isometry group over a non-archimedean field, and let cuspidal types be representations of compact open subgroups used to construct irreducible cuspidal representations by compact induction. Cuspidal-type conjugacy conjecture. Two cuspidal types are conjugate if and only if their compactly induced representations are isomorphic. This is the expected uniqueness statement for cuspidal types; the supplied text gives no resolution status.