The cumulant conjecture for spectral moments of matrix gamma distributions

Let $M\sim A\Gamma_d(\eta,\Sigma)$ be a random positive definite matrix, and let the mean spectral distribution of $M$ have moments indexed by $p$. Suppose that the asymptotic moment relation in the preceding lemma holds as $d\to\infty$ and $d/\eta\to\lambda>0$, with the right-hand side of the lemma's special case given by

$$\lim_{d\to\infty}\frac{1}{d}\int_{X>0}\left[\operatorname{tr}\left(\frac{X}{d}\right)^p\right]g_{\eta,\mathrm{I}_d}(X)\,\mathrm{d}X=\begin{cases}\lambda,&p=1,\\0,&p\geq2. \end{cases}$$

Cumulant conjecture. The right-hand side of this relation must be related to the $p$th cumulant of the $p$th moment of the mean spectral distribution of $M$, and this relation should allow identification of the limiting spectral distribution. The conjecture is presented as a proposed next step rather than a proved identification of the limiting law; the source does not provide the precise relation or determine that distribution.