The quantitative Lie–Luck conjecture for representation ranks

Let $\mathsf{G}$ be a semisimple algebraic group over $\mathbb{C}$, let $\Gamma$ be a finitely generated subgroup of $\mathsf{G}(\mathbb{C})$, fix a central character $\chi$, and let $\mathsf{X}(\mathsf{T})_\chi$ be the corresponding set of dominant integral weights. For $\bm\lambda\in\mathsf{X}(\mathsf{T})_\chi$, let $\operatorname{rk}^{\Gamma}_{\mathsf{W}_{\bm\lambda}}$ be the representation-theoretic rank function and let $\operatorname{rk}^{\chi}_\Gamma$ be the twisted von Neumann rank. Quantitative Lie–Luck conjecture. For every matrix $A$ over $\mathbb{C}[\Gamma]$,

$$\left|\operatorname{rk}^{\Gamma}_{\mathsf{W}_{\bm\lambda}}(A)-\operatorname{rk}^{\chi}_\Gamma(A)\right|=O\left(\frac{1}{\min\{\lambda_1,\ldots,\lambda_n\}}\right).$$

This predicts an explicit convergence rate for the rank functions and remains open in the generality stated.