Boundedness-by-one conjecture for ratios of minors

Let $A$ be a totally positive element of $GL_n$, and let

$$R=\frac{\text{det}(A_{I_1,I'_1})\text{det}(A_{I_2,I'_2})\cdots\text{det}(A_{I_p,I'_p})}{\text{det}(A_{J_1,J'_1})\text{det}(A_{J_2,J'_2})\cdots\text{det}(A_{J_q,J'_q})},$$

where $I_k,I'_k,J_k,J'_k\subseteq\{1,2,\ldots,n\}$ satisfy $|I_k|=|I'_k|$ and $|J_k|=|J'_k|$. A ratio is subtraction free when, in a weighted planar-network parametrization by face weights, its denominator minus numerator is a polynomial in the face weights with all coefficients positive. Boundedness-by-one conjecture. The ratio $R$ is bounded on the totally positive locus if and only if it is bounded by $1$. This conjecture asserts a sharp normalization for bounded multiplicative determinantal inequalities. Its status is not resolved by the supplied text, which explicitly says that other bounded-ratio conjectures remain open.