Strong shift equivalence and the Generalized Spectral Conjecture for nonnegative matrices

Given matrices A and B shift equivalent over a dense subring R of R, with A primitive, we show that B is strong shift equivalent over R to a primitive matrix. This result shows that the weak form of the Generalized Spectral Conjecture for primitive matrices implies the strong form. The foundation of this work is the recent result that for any ring R, the group NK1(R) of algebraic K-theory classifies the refinement of shift equivalence by strong shift equivalence for matrices over R.