Degree bound for entries of Kruithof limits

Let $A$ be a positive $m \times n$ matrix, let $V$ be a positive $m \times 1$ matrix, and let $W$ be a positive $1 \times n$ matrix whose entry sums agree. Let $x$ be the top-left entry of the Kruithof limit of $A$ with target row sums $V$ and target column sums $W$. Kruithof degree conjecture. The number $x$ is algebraic over the field generated by the entries of $A$, $V$, and $W$, with degree at most

$$\binom{m+n-2}{m-1}.$$

The conjecture is motivated by an explicit degree-$20$ computation and by the specialization from Kruithof limits to Sinkhorn limits; no general polynomial equation is known.