Factorization conjecture for limits of nonlinear random matrix statistics

Let $\lambda=(\lambda_1,\dots,\lambda_M)$ be a partition, and let $\mathcal{L}_\lambda$ denote the large-$N$ limit associated with the statistic indexed by $\lambda$. Factorization of limits. One has

$$\mathcal{L}_\lambda=\prod_{j=1}^M\mathcal{L}_{[\lambda_j]}.$$

This factorization is suggested by numerical evidence and examples, but the source says that it will be analyzed in further detail in a forthcoming publication.