Integrality conjecture for complex Wishart cumulant moments

Let $W=(NQ)^{-1}$ be the Wishart matrix described above, and define $c_0(\mu_1,\dots,\mu_\ell)$ by

$$\lim_{N\to\infty}(\beta N^2)^{\ell-1}C_\ell\left(\operatorname{tr} W^{-\mu_1},\dots,\operatorname{tr} W^{-\mu_\ell}\right)=c_0(\mu_1,\dots,\mu_\ell).$$

Here $\ell\geq 1$, $(\mu_1,\dots,\mu_\ell)\in\mathbb{N}^{\ell}$, and $C_\ell$ denotes the joint cumulant while $\operatorname{tr}=N^{-1}\operatorname{Tr}$ is the normalized trace. Integrality conjecture. For all $\ell\geq1$ and $(\mu_1,\dots,\mu_\ell)\in\mathbb{N}^{\ell}$,

$$c_0(\mu_1,\dots,\mu_\ell)\in\mathbb{N}.$$

This conjecture arises from extensive computations of the leading large-$N$ cumulants of inverse Wishart moments and predicts that these limiting quantities are always positive integers. The source does not state a resolution, so the conjecture remains open.