General polynomial equation for entries of square Sinkhorn limits

For $n \geq 1$, let $D(n,n)$ be the set of minor specifications described in the paper, and let $M(S)$ be the associated product of minors of a positive $n \times n$ matrix $A$. Let $x$ be the top-left entry of $\operatorname{Sink}(A)$. Square-matrix polynomial conjecture. There exist integers $c_S(n)$, indexed by subsets $S \subseteq D(n,n)$, such that

$$\sum_{S \subseteq D(n,n)} c_S(n)M(S)x^{|S|}=0.$$

Equivalently, the coefficient of $x^k$ is a linear combination of the $M(S)$ with $|S|=k$, and the equation has degree at most $\binom{2n-2}{n-1}$. The claim is inferred from the proven $3 \times 3$ case and interpolation for $4 \times 4$ matrices; it remains open in general.