The no-other-contributions conjecture for orbifold Gromov–Witten invariants of $\mathbb{P}^1_{3,6,6}$

Let $h_8(q)$ and $h_9(q)$ denote the generating functions for the orbifold insertions described above, and let $F_{0,3}$ and $F_{1,3}$ be the corresponding decompositions by the norm $N=a^2-ab+b^2$ modulo $3$. Let $f^{\mathbb{P}^1_{3,3,3}}_0$ and $f^{\mathbb{P}^1_{3,3,3}}_1$ be the associated generating functions for $\mathbb{P}^1_{3,3,3}$. The rhombi are counted modulo the sixfold rotational symmetry, so their degrees contribute through $q^2$. No-other-contributions conjecture. There are no contributions from $\mathbb{P}^1_{3,6,6}$ other than these rhombi; equivalently,

$$h_8(q)=\frac{1}{6}F_{0,3}(q^2)=\frac{1}{2}f^{\mathbb{P}^1_{3,3,3}}_1(q^2)$$ $${}=\frac{1}{6}+q^6+q^{18}+q^{24}+2q^{42}+q^{54}+q^{72}+2q^{78}+O(q^{96}),$$

and

$$h_9(q)=\frac{1}{6}F_{1,3}(q^2)=f^{\mathbb{P}^1_{3,3,3}}_0(q^2)$$ $${}=q^2+q^8+2q^{14}+2q^{26}+q^{32}+2q^{38}+O(q^{48}).$$

This conjecture asserts that the displayed rhombi exhaust the contributions to these quantum cohomology generating functions; the stated identities would follow once the absence of any other holomorphic orbi-sphere contributions is proved.