The strong spectral gap conjecture for higher-rank lattices

Let $\Gamma$ be a center-free irreducible higher-rank lattice in $G$, and let $V$ be a unitary representation of $\Gamma$ whose action does not extend to a $G$-action on any non-trivial subrepresentation. The strong spectral gap conjecture. The unitary induction of $V$ to $G$ has a spectral gap with respect to each factor of $G$. This strengthens the spectral gap conjecture and is known when $G$ has property T; the source gives no resolution in general.