Primitive powers conjecture for integer matrices

Let $r\geq 2$ and let $A\in \operatorname{Mat}_r(\mathbf Z)$ be nonsingular. Define $\gcd(A-I)$ as the greatest common divisor of the entries of $A-I$, and call $A$ primitive when $\gcd(A-I)=1$. Assume that $A$ has a pair of multiplicatively independent eigenvalues. Primitive powers conjecture. The matrix $A^k$ is primitive for infinitely many positive integers $k$. This conjecture proposes that, provided the matrix is initially primitive and has sufficient multiplicative independence among its eigenvalues, its powers return to the primitive condition infinitely often. The paper presents this as an analogue of the scalar conjecture for common divisors; its resolution is not given in the source.