Takahashi's higher-genus correspondence for relative invariants of $(\mathbb P^2,E)$

For every $d\in\mathbb Z_{>0}$, let $k\in\mathbb Z_{>0}$ divide $d$, and let $N_{g,d}^{\mathbb P^2/E,k}$ denote the higher-genus relative Gromov–Witten invariant associated with degree $d$, genus $g$, and contact class $k$. Here $\hbar$ is a formal variable, and the sums are taken in the formal power-series ring $\mathbb Q[\![\hbar]\!]$. Takahashi's higher-genus conjecture. For every $d\in\mathbb Z_{>0}$ and $k\in\mathbb Z_{>0}$ dividing $d$, one has

$$(-1)^{d-1}\sum_{g\geqslant 0}N_{g,d}^{\mathbb P^2/E,k}\hbar^{2g-1}=\sum_{\substack{d'\in\mathbb Z_{\geqslant 1}\k|d'|d}}\frac{1}{d/d'}(-1)^{d'-1}\sum_{g\geqslant 0}N_{g,d'}^{\mathbb P^2/E,d'}((d/d')\hbar)^{2g-1}.$$

This is proposed as a higher-genus version of Takahashi's correspondence, relating relative Gromov–Witten invariants with different divisibility data. The source presents it as conjectural; no resolution is supplied in the given text.