Non-iid Gaussian comparison conjecture for minimum eigenvalues

Let

$$\bm{Y}=\sum_{i=1}^n \bm{W}_i,$$

where the positive-semidefinite random matrices $\bm{W}_i$ are independent, and let

$$\bm{Z}=\sum_{i=1}^n \bm{X}_i,$$

where the independent random matrices $\bm{X}_i$ satisfy

$$\bm{X}_i\sim \textsc{normal}(\operatorname{\mathbb{E}}[\bm{W}_i],\ \mathsf{Mom}[\bm{W}_i]).$$

Here $\mathsf{Mom}[\bm{W}_i]$ denotes the second-moment parameter used for the Gaussian comparison model. Non-iid Gaussian comparison conjecture. Similar comparison statements should hold for the minimum eigenvalues of $\bm{Y}$ and $\bm{Z}$, including the expectation and lower-tail bounds analogous to the established iid comparison theorem. The paper establishes only weak variants of these bounds in the non-identically distributed setting. Establishing the corresponding comparison inequalities would extend the iid result to independent summands with different distributions and could yield sharper minimum-eigenvalue estimates for random positive-semidefinite matrices.