The Q-graded Kontsevich–Penner matrix-model conjecture for open intersection numbers

Let $\mathcal{M}_{(g,b),k,l}$ be the moduli space of Riemann surfaces with boundary of genus $g$, with $b$ boundary components, $k$ boundary marked points, and $l$ interior marked points, and let $\widebar{\mathcal{M}}_{(g,b),k,l}$ be its compactification by nodal surfaces. Let $F_{(g,b),n}$ denote the generating function of descendant integrals over this compactification, and let $\{T_d\}$ and $\{S_f\}$ be related to the KP times by $T_d=(2d+1)!!t_{2d+1}$ and $S_f=2^{f+1}(f+1)!t_{2f+2}$. For a matrix $\Lambda$, define $t_k(\Lambda)=\frac{1}{k}\operatorname{Tr}\Lambda^{-k}$. Q-graded Kontsevich–Penner conjecture. Assuming that the constructions of these moduli spaces and intersection integrals can be made rigorous, one has

$$\tau_Q(t_1(\Lambda),t_2(\Lambda),\ldots)=\exp\Bigl(\sum_{g,b,n}\hbar^{2g+b-2}Q^bF_{(g,b),n}\Bigr).$$

The parameter $Q$ is intended to distinguish contributions from surfaces with different numbers of boundary components. The source describes this as a conjectural refinement suggested by topological recursion and gives no resolution status; the claim is conditional on making the underlying constructions rigorous.