Lee–Pandharipande–Xiao–Xu two-point vanishing and Hodge identity

Let $X$ be the target, let $T_a$ and $T^a$ be dual cohomology bases, let $g$ be a genus, and let $B_{2g}$ be the Bernoulli number. Two-point universal-equations conjecture. For $k>g$,

$$\sum_{j=0}^{2k}(-1)^j\left\langle\left\langle\tau_j(T_a)\tau_{2k-j}(T^a)\right\rangle\right\rangle_g^X=0.$$

In addition,

$$\frac12\sum_{j=0}^{2g-2}(-1)^j\left\langle\left\langle\tau_j(T_a)\tau_{2g-2-j}(T^a)\right\rangle\right\rangle_{g-1}=\frac{(2g)!}{B_{2g}}\left\langle\left\langle\operatorname{ch}_{2g-1}(\mathbb E)\right\rangle\right\rangle_g.$$

These identities are presented as further universal equations for Gromov–Witten theory; the supplied text gives no resolution evidence.