The quasi-modularity conjecture for Hodge products on an elliptic curve

Let $E$ be an elliptic curve, let $\overline{M}_{g,n}$ be the moduli space of stable curves, and let $\mathcal C_g(\gamma_1,\ldots,\gamma_n)$ denote the class defined in the paper for insertions $\gamma_1,\ldots,\gamma_n\in H^*(E)$. Let $\lambda_{g-1}$ be the $(g-1)$st Chern class of the rank-$g$ Hodge bundle. Write $\mathrm{QMod}^E$ for the $\mathbb C$-linear span of all derivatives

$$\left(q\frac{d}{dq}\right)^iG_k(q),$$

where $i\geq 0$ and $k\geq 2$ is even. The quasi-modularity conjecture. For any $2g-2+n>0$ and $\gamma_1,\ldots,\gamma_n\in H^*(E)$,

$$\mathcal C_g(\gamma_1,\ldots,\gamma_n)\cdot\lambda_{g-1}\in H^*(\overline M_{g,n})\otimes\mathrm{QMod}^E.$$

This predicts that the Hodge-product class has coefficients in the space generated by derivatives of Eisenstein series, restricting the quasi-modular forms that can occur. The supplied text says that the authors give evidence for the conjecture but provides no resolution status.