Undecidability of testing whether the joint spectral radius is at least one

Let $\Sigma$ be a finite set of matrices, and let the joint spectral radius $\rho(\Sigma)$ be defined by

$$\rho(\Sigma)=\lim_{n\to\infty}\sqrt[n]{\max_{M_1,\dots,M_n\in\Sigma}\|M_1\cdots M_n\|}.$$

Undecidability conjecture. It is undecidable to check whether $\rho\geq 1$ for the joint spectral radius $\rho$.

The paper notes that the corresponding questions for $\rho\leq 1$ and $\rho=1$ are undecidable, whereas the threshold $\rho\geq 1$ remains open. This is relevant to the decidability of growth rates of bilinear maps.