Refined topological recursion conjecture for the multidifferentials \omega_{g,n}

Let $g\in\frac12\mathbb{Z}_{\geq0}$ and $n\in\mathbb{Z}_{\geq1}$ satisfy $2g-2+n\geq1$, and let $\omega_{g,n}$ be the multidifferential defined by Definition RTR. Refined topological recursion conjecture. The following properties hold:

• $\omega_{g,n}$ may be a divergent series, but it becomes convergent after termwise inverse Laplace transform.
• Viewed as a possibly divergent series, $\omega_{g,n}$ is a symmetric multidifferential.
• $\omega_{g,n}$ has no residues with respect to any variable.
• $\omega_{g,n}$ can have poles only at $z_i=-z_j$ and at $z_i=-\frac{k}{2}$, for all $i,j\in[n]$ and $k\in\mathbb{Z}_{\geq1}$.

These expected properties extend features of Eynard–Orantin and refined topological recursion to the non-compact spectral curve in this setting. Their proof requires controlling residues at infinitely many points and convergence of the resulting series, issues that the authors leave for future work because of substantial technical difficulties.