Effective finiteness conjecture for rational matrix families

Let $\mathcal F$ be any finite set of square matrices with rational entries, and let $\overline\rho_k(\mathcal F)$ denote the maximum normalized spectral radius among products of length $k$. Effective finiteness conjecture. There exists an effectively computable natural number $t(\mathcal F)$ such that

$$\overline\rho_{t(\mathcal F)}(\mathcal F)=\rho(\mathcal F).$$

The source states that Blondel and Tsitsiklis proved this conjecture false, so no such effective bound exists in the asserted generality.