Rectangular Sinkhorn-limit polynomial conjecture with sign alterations

Let $A$ be a positive $m \times n$ matrix, let $D(m,n)$ be the set of minor specifications described in the paper, and let $M(S)$ be the associated product of minors. For $S \subseteq D(m,n)$, let $\operatorname{adj}_{S,\sigma(S)}(m,n)$ be the sign-altered adjacency matrix defined in the paper. Let $x$ be the top-left entry of $\operatorname{Sink}(A)$. Rectangular polynomial conjecture. For every $S \subseteq D(m,n)$, there is a sign alteration $\sigma(S)$, with diagonal signs equal to $1$, such that

$$\sum_{S \subseteq D(m,n)}\left(\det\operatorname{adj}_{S,\sigma(S)}(m,n)\right)M(S)x^{|S|}=0.$$

The sign alterations can be chosen invariant on equivalence classes, independent of $m$ and $n$, and dependent only on the link structure of $S$. This is the paper's main open question; identifying canonical correct sign alterations remains unresolved.