The quantitative normalized cohomology 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 $\lambda_i=\lambda_i(k)$ be the highest weight parameters of irreducible representations $\mathsf{W}_k$. Quantitative normalized cohomology conjecture. For every $i$,

$$\left|\frac{\dim \operatorname{H}^i(\Gamma,\mathsf{W}_k)}{\dim \mathsf{W}_k}-b_i^{(2)}(\Gamma)\right|=O\left(\frac{1}{\min\{\lambda_1,\ldots,\lambda_n\}}\right).$$

This strengthens the qualitative convergence claim by predicting an explicit rate of convergence. The estimate is established only in special cases in the paper, so it remains open in the stated generality.