Positive-integrality conjecture for time-delay matrix cumulants

Let $X_n$ be distributed according to the Laguerre ensemble with parameter $\alpha=n$, and define the rescaled inverse power traces

$$\tau_k(n)=n^{k-1}\operatorname{Tr}X_n^{-k}\qquad (k\geq 0).$$

Their expectations have the expansion

$$\mathbb{E}\tau_k(n)=\sum_{g=0}^{\infty}\kappa_g(k)n^{-g}.$$

Positive-integrality conjecture. For $\beta\in\{1,2\}$, the expansion coefficients satisfy

$$\kappa_g(k)\in\mathbb{N}.$$

This conjecture concerns the $1/n$-expansion of cumulants and moments of the time-delay matrix for ballistic chaotic cavities, where the inverse power traces are singular spectral statistics of the Laguerre ensemble. The source gives no resolution, so the conjecture remains open.