Enumerative interpretation of Gromov–Witten invariants for the cubic degeneration and [?]2

Let $X$ be either the Hirzebruch surface $\mathbb F_2$ or the surface $B'$ obtained by blowing up six points on a conic, and let $E$ be its $(-2)$-curve. Let $D$ be an effective divisor class on $X$ with $D\ne 0$ and $D\ne E$, and let $\gamma$ denote the class of a point. A stable map is understood as a map $\pi:C\to X$ from a curve $C$, with pushforward class $\pi_*[C]=D$. Enumerative Gromov–Witten conjecture. The Gromov–Witten invariant

$$I_{g,D}\left(\gamma^{-K_X\cdot D+g-1}\right)$$

is the number of maps $\pi:C\to X$ such that $C$ has one component $C_0$ not mapping to $E$, every other component $C'$ maps isomorphically to $E$, and $C'$ intersects $\overline{C\setminus C'}$ at one point contained in $C_0$. This conjectural interpretation would extend the recursive computation of Gromov–Witten invariants from del Pezzo surfaces to the singular degeneration $B'$ and, by deformation invariance, to cubic surfaces. The corresponding genus-zero interpretation for $\mathbb F_2$ was attributed to Kleiman and Abramovich; the general assertion in the stated form is not resolved by the supplied source.