Takahashi's logarithmic mirror symmetry conjecture for plane curves

Let $E\subset\mathbb P^{2}$ be an elliptic curve with a chosen group structure whose zero element is a flex point, and let $P\in E$ be a point of order $3d$. Let $m_{d}$ be the number of rational degree $d$ curves in $\mathbb P^{2}$ meeting $E$ only at $P$ and along only one branch, and let $n_{d}$ be the local BPS state count of $\mathbb P^{2}$ in degree $d$. Takahashi's conjecture.

$$3d\,m_{d}=(-1)^{d+1}n_{d}.$$

This conjecture proposes a direct logarithmic mirror-symmetry relation between the enumerative counts $m_d$ and the local BPS counts $n_d$, motivated by Takahashi's calculations and related work of Gathmann.