Boyle–Handelman spectral conjecture for primitive matrices over subrings of the reals

Let $\mathcal R$ be a subring of $\mathbb R$, and let $\Lambda=(\lambda_1,\dots,\lambda_k)$ be a $k$-tuple of complex numbers. The necessary conditions are the Perron condition, the coefficients condition, and the relevant trace conditions stated in the source, including nonnegative traces and, when $\mathcal R=\mathbb Z$, nonnegative net traces. Boyle–Handelman's spectral conjecture. The tuple $\Lambda$ is the nonzero spectrum of some primitive matrix over $\mathcal R$ if and only if it satisfies those necessary conditions. This is a realization conjecture for the stable nonnegative inverse eigenvalue problem; the source presents it as a conjecture without supplying a resolution.