Joint tracial moment conjecture for the limiting spectral distribution of data-matrix commutators

Let $\Sigma_1$ and $\Sigma_2$ be the covariance matrices, and let $H$ be their joint limiting spectral distribution. For any $n_1,n_2\in\mathbb{N}$ and every selection of non-negative integers $(k_1,\ldots,k_{n_1})$ and $(l_1,\ldots,l_{n_2})$, consider the alternating joint tracial moments of $\Sigma_1$ and $\Sigma_2$. Joint tracial moment conjecture. Suppose that

$$\frac{1}{p}\operatorname{trace}(\Sigma_1^{k_1}\Sigma_2^{l_1}\Sigma_1^{k_2}\Sigma_2^{l_2}\ldots) \longrightarrow \int \lambda_1^{\sum_i k_i}\lambda_2^{\sum_j l_j}\,dH(\lambda_1,\lambda_2).$$

In other words, the joint tracial moments converge to quantities determined by the joint limiting spectral distribution of $(\Sigma_1,\Sigma_2)$. Then the conclusion of the paper's main theorem on the limiting spectral distribution holds. This is proposed as a sufficient condition for the main result to remain valid without the commutativity assumption.