Breuillard's uniform exponential growth conjecture for linear groups

Given $d\in\mathbb{N}$, let $S$ be a finite subset of $\operatorname{GL}_d(\mathbb{C})$ and define its exponential growth rate by

$$\rho_S:=\lim_{n\to+\infty}\frac{1}{n}\log|S^n|,$$

where $S^n$ is the set of products of $n$ elements of $S$. Breuillard's uniform exponential growth conjecture. For every $d\in\mathbb{N}$, there is $\varepsilon=\varepsilon(d)>0$ such that, for every finite subset $S\subseteq\operatorname{GL}_d(\mathbb{C})$, either $\rho_S=0$ or $\rho_S>\varepsilon$.

Examples of Grigorchuk and de la Harpe show that exponential growth rates can be arbitrarily small and positive even for linear groups; the conjecture asserts that, in each fixed dimension, positive exponential growth is uniformly bounded away from zero. The supplied source does not indicate whether this conjecture has been resolved.