Setup: Remark 5.1 in this work states that when $X\in\mathbb{R}^{n\times p}$ have i.i.d. $N(0,1/p)$ entries, the limit distribution of $XX^\top$ is the Marcenko-Pastur law, with limiting rectangular free cumulants $$ \kappa_2^\infty=1 \text{ and } \kappa_{2j}^\infty=0 \text{ for all $j\geq 2$,} $$ as $n,p\rightarrow\infty$ and $\frac{n}{p}\rightarrow\delta$, where $\delta$ is some non-growing constant.
Question: What happens to the limiting rectangular free cumulants of the matrix $X$ above if we swap the variance of its entries to be $\frac{1}{p}\times CW_{ij}$ (instead of $\frac{1}{p}$), where $CW_{ij}$ is a non-growing constant as $n,p\rightarrow\infty$. In other words, $X$ has independent $N\big(0,\frac{1}{p}\times CW_{ij}\big)$ entries. Thanks.
For clarification: We now have $X=(X_{ij})$ where each matrix entry $X_ij$ is now normally distributed with variance $\frac{1}{p}×CW_{ij}$. In particular, the entries of $X$ are still independent but no longer identically distributed due to their dependence on $i$ and $j$ through $W_{ij}$.