Mariño–Vafa conjecture for triple Hodge integrals

For the Deligne–Mumford moduli space of stable curves $\overline{{\mathcal M}}_{g,h}$, let $\Lambda_g^\vee(u)$ denote the total Chern polynomial of the Hodge bundle and let $\psi_i$ be the cotangent-line classes. The triple Hodge integrals are

$$\int_{\overline{{\mathcal M}}_{g,h}}\frac{\Lambda_g^\vee(1)\Lambda_g^\vee(\tau)\Lambda_g^\vee(-\tau-1)}{\prod_{i=1}^h(1-\mu_i\psi_i)}.$$

Let $G(\lambda;\tau;p)$ be the generating series of these integrals and $R(\lambda;\tau;p)$ the representation-theoretic generating series defined using symmetric-group characters and the quantities ${\mathcal W}_\mu(\lambda)$. Mariño–Vafa conjecture. One has the identity

$$G(\lambda;\tau;p)=R(\lambda;\tau;p).$$

The conjecture expresses all-genus, all-marked-point triple Hodge integrals through finite representation-theoretic or Chern–Simons data. In the supplied paper it is subsequently proved via matching cut-and-join equations and the initial value at $\tau=0$.