Projectivity characterization for restrictions from G to H

Let $G$ be the ambient group, let $H$ be the subgroup, and let $\pi$ be an irreducible representation of $G$. The restriction $\pi|_H$ is the representation of $H$ obtained from $\pi$ by restricting the action to $H$.

Projectivity conjecture. The restriction $\pi|_H$ is projective if and only if $\pi$ is supercuspidal.

This conjecture asks whether projectivity of the restriction to $H$ exactly characterizes supercuspidal representations of $G$. The surrounding discussion notes that restrictions of supercuspidal representations are known to be projective, while the converse is the proposed assertion; the parser supplies no evidence of resolution.