Holomorphic anomaly equation for elliptic-surface Gromov–Witten series

For $g\geq 0$ and $n\geq 1$, let $Z_{g;n}(q):=\sum_dN_{g,d;n}q^d$, where $N_{g,d;n}:=\sum_{(\beta,\sigma)=d,\,(\beta,F)=n}N_g(\beta)$. Let $E_2(q),E_4(q),E_6(q)$ be Eisenstein series of weights $2,4,6$. Holomorphic anomaly equation. The series has the form

$$Z_{g;n}(q)=\frac{P_{2g+6n-2}(E_2(q),E_4(q),E_6(q))}{\prod_{k\geq 1}(1-q^k)^{12n}},$$

where $P_{2g+6n-2}$ is homogeneous of weight $2g+6n-2$, and it satisfies

$$\frac{\partial P_{2g+6n-2}}{\partial E_2}=\frac{1}{24}\sum_{g=g'+g”}\sum_{s=1}^{n-1}s(n-s)P_{2g'+6s-2}P_{2g”+6(n-s)-2}+\frac{n(n+1)}{24}P_{2(g-1)+6n-2}.$$

This conjecture is a modular-form expression for the generating functions and their recursive anomaly relation, motivated by physical holomorphic-anomaly arguments; the supplied text gives no resolution status.