The normalized cohomology convergence conjecture for semisimple algebraic groups

Let $\mathsf{G}$ be a semisimple algebraic group over $\mathbb{C}$, let $\Gamma$ be a torsion-free subgroup of type $FP_\infty$, and let $\{\mathsf{W}_k\}$ be a sequence of irreducible representations of $\mathsf{G}$ whose highest weight parameters grow to infinity. Normalized cohomology convergence conjecture. For every $i$,

$$\lim_{k \to \infty} \frac{\dim \operatorname{H}^i(\Gamma, \mathsf{W}_k)}{\dim \mathsf{W}_k}=b_i^{(2)}(\Gamma),$$

where $b_i^{(2)}(\Gamma)$ is the $i$-th $\ell^2$-Betti number of $\Gamma$. This conjecture gives an asymptotic description of the cohomology of representations whose highest weights escape to infinity; the paper provides positive answers in the case of products of copies of $\operatorname{SL}_2(\mathbb{C})$, while the general case remains open.