The generalized Witten conjecture for geometric descendent potentials

Let $\mathcal C=(\Sigma,x,y)$ be a spectral curve with $m$ boundaries. Let $\mathcal D({\bf t};\hbar)$ be its geometric total descendent potential, and for $\mathfrak g(\Sigma)\geq1$ let $\mathcal D^{\rm NP}_{\mu,\nu}({\bf t};\hbar;w)$ be the non-perturbative total descendent potential. Let ${\bf t}={\bf t}({\bf p})$ be a coordinate transformation and let $Q_\Delta({\bf p},{\bf p})$ be a quadratic function. Generalization of the Witten conjecture. There exist ${\bf t}={\bf t}({\bf p})$ and $Q_\Delta({\bf p},{\bf p})$ such that, for $\mathfrak g=0$, $e^{Q_\Delta({\bf p},{\bf p})/2}\mathcal D(\hbar\cdot{\bf t}({\bf p});\hbar)$ is a tau-function of the $m$-component KP hierarchy with KP times $\{p_k^i/k\}_{k\geq1,\,1\leq i\leq m}$, while for $\mathfrak g\geq1$, $e^{Q_\Delta({\bf p},{\bf p})/2}\mathcal D^{\rm NP}_{\mu,\nu}(\hbar\cdot{\bf t}({\bf p});\hbar;w)$ is such a tau-function; both satisfy certain reduction structures. This combines the topological-recursion integrability claim with the TR–geometry correspondence. The paper presents it as a conjectural generalization; the full statement remains open.