Lian's quasimodularity conjecture for elliptic Hurwitz loci

Fix an integer $g\ge2$. Let $\overline{\mathcal{M}}_g$ be the moduli space of stable curves of genus $g$, and for each $d\ge1$ let $\phi_{g/1,d}$ be the morphism from the compactified Hurwitz space of simply branched degree-$d$ covers from genus-$g$ curves to genus-$1$ curves, remembering the source curve and its ramification points. Write $(\phi_{g/1,d})_*(1)$ for the resulting cycle class, and let $A^{g-1}(\overline{\mathcal{M}}_g)$ denote the codimension-$g-1$ Chow group. Let $\operatorname{Qmod}$ be the ring of quasimodular forms. Lian's quasimodularity conjecture. The generating series of these Hurwitz loci satisfies

$$\sum_{d\ge1}[(\phi_{g/1,d})_*(1)]q^d\in A^{g-1}(\overline{\mathcal{M}}_g)\otimes\operatorname{Qmod}.$$

The conjecture predicts that the elliptic Hurwitz loci exhibit the same quasimodular-form structure familiar from generating series of curve-counting invariants. Its status is not established by the supplied source context.