Subtraction-free characterization conjecture for bounded determinantal ratios

Let $A$ be a totally positive element of $GL_n$, and let $R$ be a ratio of products of minors as above. In a weighted planar-network parametrization, call $R=p/q$ subtraction free if $q-p$ is a polynomial in the face weights with all coefficients positive. Subtraction-free characterization conjecture. The ratio $R$ is bounded on the totally positive locus if and only if it is subtraction free. This conjecture seeks an algebraic characterization of bounded determinantal inequalities through positive face-weight expressions. The supplied text does not establish its status; it is presented as an open conjecture in the paper's discussion.