The universal formula conjecture for the x-y swap in topological recursion

Let $g\geq 0$ and $n\geq 1$. Let $\omega^{(g)}_{0,n}$ be the $n$-differentials associated with the original variables, let $\omega^{(h)}_{m,0}$ denote the corresponding differentials with $m$ arguments, and let $x_i,y_i$ be the paired variables. Write $\mathsf{Expr}_{g,n}$ for the universal closed differential-algebraic expression in these differentials and in the forms $dx_i/x_i$, $dy_i/y_i$, and $dx_i dx_j/(x_i-x_j)^2$. Universal formula conjecture. For $g\geq 0$ and $n\geq 1$, one has

$$\omega^{(g)}_{0,n}=\mathsf{Expr}_{g,n}\Big(\big\{\omega^{(h)}_{m,0}\big\}_{2h-2+m\leq 2g-2+n},\Big\{\frac{dx_i}{x_i},\frac{dy_i}{y_i}\Big\}_{i=1,\dots,n},\Big\{\frac{dx_i dx_j}{(x_i-x_j)^2}\Big\}_{\substack{i,j=1,\dots,n\\ i\ne j}}\Big).$$

The conjecture seeks an explicit universal expression for the $x$-$y$ swap in topological recursion. The surrounding discussion states that related universal expressions exist and are used in enumerative geometry, matrix models, and free probability, while the particular expression is presented as not yet explicitly available.