Takahashi's higher-genus BPS invariance conjecture for $(\mathbb P^2,E)$

For every $d\in\mathbb Z_{>0}$ and every positive divisor $k$ of $d$, let $\Omega_{d,k}^{\mathbb P^2/E}(\hbar)\in\mathbb Q[\![\hbar]\!]$ be the higher-genus relative BPS invariant defined from the relative invariants $N_{g,d}^{\mathbb P^2/E,k}$. Let $k'$ be another positive divisor of $d$. Takahashi's higher-genus BPS conjecture. For every $d\in\mathbb Z_{>0}$ and every pair of positive divisors $k,k'$ of $d$, one has

$$\Omega_{d,k}^{\mathbb P^2/E}(\hbar)=\Omega_{d,k'}^{\mathbb P^2/E}(\hbar).$$

Equivalently, the higher-genus relative BPS invariant depends on $d$ but not on the divisor $k$. The source states this as a reformulation of the preceding conjecture, and no resolution is supplied in the given text.