Shalom's 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 $\Gamma$. Assume that the $\Gamma$-action on $V$ does not extend to a $G$-action on any non-trivial subrepresentation. Shalom's spectral gap conjecture. Then $V$ has a spectral gap. The source states that this is open and that, for unitary representations, it is equivalent to the degree-$1$ case of the semisimple below-rank cohomology conjecture.