Degree-bound conjecture for recursive determination of quintic Gromov-Witten invariants

Let $S_{g,d}$ and $S'_{g,d}$ be the indexing sets used in the paper, let $B_{g,d}(\rho)$ and $C_\rho$ be the corresponding terms and coefficients in the degeneration equations, and let $N_{g,d}$ be the degree-$d$, genus-$g$ Gromov–Witten invariant of the quintic threefold $Q$. Degree-bound conjecture. For every pair $(g,d)$ satisfying

$$d\geq\frac{2g-1}{5},$$

there exists $\zeta_{g,d}\in S_{g,d}$ such that

$$B_{g,d}(\zeta_{g,d})-\sum_{\rho\in S'_{g,d}}B_{g,d}(\rho)C_\rho=C_{g,d}N_{g,d},\qquad C_{g,d}\neq0.$$

The assertion is intended to provide the nonzero equation needed to recursively determine $N_{g,d}$ beyond the cases $g=2,3$ proved in the paper; its validity for all pairs under the stated degree bound remains open.