Pole structure conjecture for generalized BGW correlation differentials

For stable pairs $(g,n)$ with $2g+n-2>0$, let $\omega_{g,n}$ be the differentials defined from the correlation functions $W_{g,n}$ on the spectral curve. They are symmetric in the variables $z_j$. For any $g$ and $n$, the quotient

$$\frac{z_1^2\dots z_n^2\,\omega_{g,n}(z_1,\dots,z_n)}{d z_1\dots dz_n}$$

is a polynomial in each of the variables $z_1^{-1},\dots,z_n^{-1}$.

Pole structure conjecture. All $\omega_{g,n}$ are meromorphic differentials on the spectral curve and have poles of finite degree only at the branch point $z_j=0$.

This conjecture describes the expected pole structure needed for a Chekhov--Eynard--Orantin topological-recursion treatment. The source suggests that it should be provable by topological-recursion methods, but does not establish it in the paper.