Topological recursion for higher BGW tau-functions

Let $m$ be a positive integer, and let the irregular spectral curve be given by

$$(m+1)xy^{m+1}=1.$$

The higher BGW tau-functions are denoted by $\tau^{(m)}$.

Higher BGW topological-recursion conjecture. The generating functions of the higher BGW models $\tau^{(m)}$ are given by a version of the Chekhov--Eynard--Orantin topological recursion on the irregular spectral curve $(m+1)xy^{m+1}=1$, with parameters $r=m+1$ and $s=m$.

This identifies the higher BGW generating functions with topological-recursion invariants of the corresponding irregular spectral curve. The source attributes this description to a version of the recursion discussed by Borot, but the supplied text does not establish whether the statement is proved or remains conjectural.