Fallat–Gekhtman–Johnson conjecture on factorization of bounded ratios of minors

Let $A$ be a totally positive $n\times n$ matrix. For index sequences $\alpha$ and $\beta$ defining products of minors, write $\alpha/\beta$ for the corresponding ratio. A ratio is subtraction-free if, when expressed as a rational function $p/q$ in the face weights, the polynomial $q-p$ has all positive coefficients. A primitive ratio is one of the primitive ratios defined for ratios of minors of totally positive matrices. Fallat–Gekhtman–Johnson's conjecture. Let $\alpha/\beta$ be a ratio. Then (1) $\alpha/\beta$ is bounded on all totally positive matrices if and only if it can be written as a product of primitive ratios; and (2) $\alpha/\beta$ is bounded on all totally positive matrices if and only if it is subtraction-free. The paper gives counterexamples to the factorization assertion, while reporting that all found examples satisfy the subtraction-free assertion; hence the combined conjecture is not established as stated.