Laurent-polynomial conjecture for membrane indices on $\mathcal{O}(-2)\oplus\mathcal{O}(2)$ over $\mathbb{P}^3$

Let $Z$ be the local Calabi–Yau fivefold in the example over $\mathbb{P}^3$, let $[L]$ be the class of a line in $\mathbb{P}^3$, and let $q_0,q_1,q_2,q_3$ be the associated equivariant variables. Denote by $\Omega_{d[L]}$ the membrane index in degree $d[L]$.

Laurent-polynomial conjecture. For every degree $d$, $\Omega_{d[L]}$ is a Laurent polynomial in $q_0$, $q_1$, $q_2$, and $q_3$.

The conjecture predicts equivariant Laurent-polynomiality of the membrane indices in this local example. The source reports experimental evidence from low-degree calculations but does not establish the claim in general.