The m-KP integrability conjecture of topological recursion

Let $\mathcal C=(\Sigma,x,y)$ be a spectral curve with boundaries ${\bf b}=\{b_1,\ldots,b_m\}$, and let $\Lambda=(\lambda_1,\ldots,\lambda_m)$ be local coordinates near the boundaries. Write $\mathfrak g=\mathfrak g(\Sigma)$, and let $Z({\bf p};\hbar)$ be the generating series of topological-recursion descendents. For $\mathfrak g\geq 1$, let $Z^{\rm NP}_{\mu,\nu}({\bf p};\hbar;w)$ be its non-perturbative version. $m$-KP integrability conjecture. If $\mathfrak g=0$, then $Z({\bf p};\hbar)$ is a tau-function of the $m$-component KP hierarchy with KP times $\{p_k^i/k\}_{k\geq1,\,1\leq i\leq m}$. If $\mathfrak g\geq1$, then $Z^{\rm NP}_{\mu,\nu}({\bf p};\hbar;w)$ is a tau-function of the same hierarchy with those KP times. This is the integrability part of the proposed generalization of the Witten conjecture. The paper proves the integrability conjecture for one-boundary cases, but the general claim remains open.