The conjecture on geometrically minimal K-types

Let $(\pi,V)$ be an irreducible supercuspidal representation of $G$, and let ${\mathscr{S}}$ be the $G$-equivariant Schneider–Stuhler sheaf on the building $X$ associated with $(\pi,V)$. Let $B$ be a minimal ball in $X$ supporting a section of ${\mathscr{S}}$, let $\sigma$ be the center of $B$, and set $K=G_\sigma$. The space $H_B^0(X,{\mathscr{S}})$ of sections supported on $B$ is called a geometrically minimal $K$-type. The geometrically minimal K-type conjecture. The space $H_B^0(X,{\mathscr{S}})$ is irreducible and

$$(\pi,V)=\operatorname{cInd}_K^G\bigl(H_B^0(X,{\mathscr{S}})\bigr).$$

It is also unique up to conjugation: if $B'$ is another minimal ball supporting a section of ${\mathscr{S}}$, then there exists $g\in G$ such that $gB=B'$. This conjecture refines the compact induction conjecture by identifying the inducing data geometrically; its status in the general higher-rank setting is open.