The collapse conjecture for Schneider–Stuhler sheaves

Let $X$ be the building of $G$, and let ${\mathscr{S}}$ be the Schneider–Stuhler $G$-equivariant sheaf associated with an irreducible supercuspidal representation. A basic elementary sheaf supported on adjacent facets $\{\sigma,\tau\}$ has one-dimensional stalks at $\sigma$ and $\tau$ and an isomorphic restriction map; an elementary sheaf is a direct sum of such basic elementary sheaves. We say that ${\mathscr{S}}$ collapses equivariantly onto ${\mathscr{S}}'$ if an equivariant elementary subsheaf ${\mathscr{J}}\subset{ \mathscr{S}}$ \exists with ${\mathscr{S}}'\cong{\mathscr{S}}/{\mathscr{J}}$. The collapse conjecture. The sheaf ${\mathscr{S}}$ can be equivariantly collapsed, through a finite sequence of equivariant collapses, onto an indecomposable equivariant injective sheaf. Under finiteness conditions, such injective sheaves are direct sums of constant sheaves supported on closures of facets, so the conjecture predicts a particularly explicit model for ${\mathscr{S}}$. Its general status is open.