Hadamard product of independent random sample covariance matrices with correlation structure

We compute the asymptotic empirical eigenvalue distribution of the matrix $M={⨀}_{i=1}^k\frac{1}{d_i}{X}^{(i)}{X^{(i)}}^{\mathrm{\top }}$ where $X^{(i)}\in {\mathbb{R}}^{n\times d_i}$ are independent matrices with independent rows but general correlation within each row under the dimension scaling $\frac{n}{d_1\dots d_k}\to \gamma$.