Liu–Xu Hodge-integral conjecture

Let $g\geq2$, let $n$ be a positive integer, and let $d_j\geq1$ satisfy

$$\sum_{j=1}^{n}(d_j-1)=g.$$

Let $\lambda_i$ denote the Chern classes of the Hodge bundle, let $\tau_d$ denote a psi-class insertion, let $B_{2g-2}$ be the Bernoulli number, and write $\langle\prod_{j=1}^n\tau_{d_j}\mid\lambda_a\lambda_b\rangle_g$ for the corresponding Hodge integral. Liu–Xu's Hodge-integral conjecture.

$$\frac{2g-2}{|B_{2g-2}|}\left(\left\langle\prod_{j=1}^n\tau_{d_j}\mid\lambda_{g-1}\lambda_{g-2}\right\rangle_g-3\left\langle\prod_{j=1}^n\tau_{d_j}\mid\lambda_{g-3}\lambda_g\right\rangle_g\right)=\frac{1}{2}\sum_{j=0}^{2g-4}(-1)^j\left\langle\tau_{2g-4-j}\tau_j\prod_{i=1}^n\tau_{d_i}\right\rangle_{g-1}+\frac{(2g-3+n)!}{2^{2g+1}(2g-3)!\prod_{j=1}^n(2d_j-1)!!}.$$

The source presents this as an identity equivalent to the preceding conjectural psi-intersection formula, using the relation between $\operatorname{ch}_{2g-3}(\mathbb E)$ and Hodge classes; no resolution is given in the supplied text.