Cuspidality criterion in the refined local Langlands correspondence

Fix a reductive group $G$, a rigid inner twist, a Whittaker datum $\mathfrak w$, and the refined local Langlands reciprocity map $\operatorname{rec}_{\mathfrak w}$. An enhanced $L$-parameter is called cuspidal according to the parameter-side cuspidality notion. Cuspidality criterion. An irreducible representation $\dot\pi$ is supercuspidal if and only if $\operatorname{rec}_{\mathfrak w}(\dot\pi)$ is cuspidal. This is the foundational compatibility expected between cuspidal enhanced $L$-parameters and supercuspidal representations.