Faber's two-point Gromov–Witten relations and Hodge-class identity

Let $X$ be a smooth projective variety and let $\{T_a\}$ be a basis of $H^*(X,\mathbb Q)$, with dual basis $T^a=g^{ab}T_b$ for the Poincaré pairing. Write $\langle\langle\cdots\rangle\rangle_g^X$ for the genus-$g$ Gromov–Witten potential, $\mathbb E$ for the Hodge bundle, and $B_{2g}$ for the Bernoulli number. Faber's conjecture. For $k>g$,

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

Moreover,

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

The paper presents these identities in the discussion of the point case and does not provide a resolution of the stated relations.