Takahashi's log mirror symmetry conjecture for the projective plane

Let $E\subseteq\mathbb{P}^2$ be a smooth cubic, and let $m_d$ denote the number of affine-line curves of degree $d$ meeting $E$ at a chosen point of order $3d$. Let $n_d$ be the local BPS state count of degree $d$ for local $\mathbb{P}^2$.

Takahashi's conjecture. For all $d\geq 1$,

$$3d\,m_d=(-1)^{d+1}n_d.$$

This is the $A$-model formulation of Takahashi's proposed log mirror symmetry for $(\mathbb{P}^2,E)$, asserting that its invariants agree with those of local $\mathbb{P}^2$ up to the displayed sign and factor. The source refers to Takahashi for the precise $B$-model statement.