Positivity conjecture for the two-variable Kerov polynomial correction

Let $r,s$ be positive integers, let $\Phi(x,j)$ be the generating-function expression used in the preceding propositions, and write $[x^{r+1}]$ and $[y^{s+1}]$ for coefficient extraction. The resulting expression is a polynomial in free cumulants. Two-variable positivity conjecture. The following quantity is positive in free cumulants:

$$[x^{r+1}] [y^{s+1}] \prod_{j=0}^{r-1} \Phi(x,j) \prod_{j=0}^{s-1} \Phi(y,j) \frac{(xy)^2}{(y - x -rxy)(x-y -sxy)}.$$

This conjecture is based on numerical evidence and concerns the positivity of the correction term in the formula for $\Sigma_{r,s}$. The source notes that positivity of the original Kerov polynomials remains open.