The non-uniform lattice extension of Gromov's $L^p$-cohomology theorem

Let $G$ be a non-compact semisimple group over a local field $F$ with finite center and rank $r$. The non-uniform lattice extension conjecture. The full statement of Gromov's $L^p$-cohomology theorem for $G$ also holds for non-uniform irreducible lattices. The theorem already establishes below-rank vanishing, Hausdorffness at the rank, suitable non-vanishing at the rank, and above-rank vanishing in the stated ranges for $G$ and with possible exceptions for non-uniform lattices; the conjecture asks that these conclusions hold without those exceptions.