The semisimple below-rank cohomology conjecture for super-reflexive coefficients

Let $G$ be an $S$-semisimple group of rank $r$ and let $\Gamma<u001b[0mG$ be an irreducible lattice, meaning that its projection to each simple factor of $G$ is dense. Let $V$ be a super-reflexive Banach space and let $\Gamma\to B(V)$ be a linear isometric representation. The semisimple below-rank cohomology conjecture. There exists a $\Gamma$-invariant subspace $V_0\subset V$ on which the $\Gamma$-action extends to a $G$-action and such that, for every $j<r$,

$$H^j(\Gamma,V)\cong H^j(\Gamma,V_0)\cong H^j(G,V_0),$$

where the left map is induced by $V_0\hookrightarrow V$ and the right map is restriction from $G$ to $\Gamma$. In particular, if the $\Gamma$-action on $V$ does not extend to a $G$-action on any non-trivial subrepresentation, then $H^j(\Gamma,V)=0$. This extends the simple-group super-reflexive property $(T_{r-1})$ conjecture to semisimple groups and predicts that all below-rank cohomology comes from the ambient group; it is explicitly open even in degree $1$.