Higher-degree denominator conjecture for membrane indices on $\mathcal{O}(-1)\oplus\mathcal{O}(3)$ over $\mathbb{P}^2$

Let $\overline{q}_0,\overline{q}_1,\overline{q}_2$ be the equivariant variables in the example of the local Calabi–Yau fivefold over $\mathbb{P}^2$, let $[H]$ denote the hyperplane class, and let $\Omega_{d[H]}$ be the corresponding membrane index. For an integer $n$, write $\operatorname{od}(n)$ for its odd part, and let $[m]_{\overline{q}_i}$ denote the quantum-number notation used for the equivariant variable $\overline{q}_i$.

Higher-degree denominator conjecture. For every integer $d>1$,

$$\left(\prod_{i=0}^2 \prod_{n=1}^{d-1}\frac{[2n]_{\overline{q}_i}}{[\operatorname{od}(n)]_{\overline{q}_i}}\right)\cdot \Omega_{d[H]} \in \mathbb{Z}[e_1^{\pm 1},e_2,e_3].$$

The claim predicts that, after multiplying by the displayed factors, higher-degree membrane indices have integral coefficients. It is motivated by numerical data and the observation that powers of $2$ are the worst denominators in the non-skeletal limit; the source gives no proof or resolution.