Boyle–Handelman strong generalized spectral conjecture

Let $A$ be a square matrix over a subring $\mathcal R$ of $\mathbb R$. Its nonzero spectrum is the multiset of its nonzero eigenvalues. Boyle–Handelman's strong generalized spectral conjecture. If the nonzero spectrum of $A$ satisfies the necessary conditions of the Spectral Conjecture, then $A$ is strong shift equivalent over $\mathcal R$ to a primitive matrix. This strengthens the weak generalized conjecture by replacing shift equivalence with strong shift equivalence; the source gives no resolution.