Blondel–Jungers conjecture for pairs of binary matrices

A binary matrix is a matrix whose entries belong to $\{0,1\}$. For a finite matrix family, the finiteness property means that some finite product attains the joint spectral radius. Blondel–Jungers conjecture. Every pair of binary matrices has the finiteness property. The source presents this as an open reduction of the rational-matrix finiteness problem and reports only the $2\times2$ case as proved.