The binary-matrix finiteness conjecture

Let $\Sigma=\{A_1,A_2\}$ be a pair of binary matrices, meaning that every entry of each matrix belongs to $\{0,1\}$. The pair has the finiteness property if there exist $t\geq 1$ and a product $A=A_{i_1}\cdots A_{i_t}$ with $A_{i_j}\in\Sigma$ such that

$$\rho(\Sigma)=\rho(A)^{1/t}.$$

Binary-matrix finiteness conjecture. Every pair of binary matrices has the finiteness property.

The conjecture is motivated by computations and applications involving binary matrices as adjacency matrices. The paper proves that finiteness for all pairs of binary matrices is equivalent to finiteness for all sets of nonnegative rational matrices; the surrounding text presents the binary assertion as an open conjecture.