The conifold gap conjecture for local $\mathbb{P}^2$ relative to an elliptic curve

Let $t_{\mathrm{con}}$ be the flat coordinate near the conifold point, and let $F_{g,\mathrm{con}}^{\mathbb{P}^2\!/\!E}$ denote the genus-$g$ free energy expressed as a function of this coordinate. The conifold point is given by $t_{\mathrm{con}}=0$. Conifold gap conjecture. For every $g\geq 2$,

$$F_{g,\mathrm{con}}^{\mathbb{P}^2\!/\!E}=-\frac{2^{2g-1}-1}{2^{2g-1}}\frac{|B_{2g}|}{2g(2g-1)(2g-2)}\frac{1}{t_{\mathrm{con}}^{2g-2}}+O(1)$$

near the conifold point, where $B_{2g}$ denotes the $2g$-th Bernoulli number. This is a proposed analogue for $F_g^{\mathbb{P}^2\!/\!E}$ of the conifold gap conjecture for local $\mathbb{P}^2$. The corresponding local $\mathbb{P}^2$ conjecture is known in low genus but remains open in general; the status of this relative version is not resolved in the supplied text.