The nonvanishing conjecture for the FJRW recursion coefficient

Let $\Theta_{g,k}$ denote the genus $g$ FJRW invariant of the Fermat quintic with $k$ insertions, and let $e_{g,k}$ be the scalar coefficient of $\Theta_{g,k}$ in the polynomial relation produced by the proposed localization algorithm.

Nonvanishing conjecture. The coefficient satisfies

$$e_{g,k}\neq0.$$

When this coefficient is nonzero, the proposed relation evaluates $\Theta_{g,k}$ from invariants of lower genus and from $\Theta_{g,k'}$ with $k'<k$. The conjecture is therefore the key missing condition for the recursive FJRW algorithm.