The rank-sensitive matrix discrepancy conjecture

Let $\xi_1,\ldots,\xi_n$ be independent scalar random variables with finite support, and let $\mathbf{A}_1,\ldots,\mathbf{A}_n\in {\mathbb C}^{d\times d}$ be Hermitian matrices satisfying ${\rm rank}(\mathbf{A}_i)\leq r$ for $i=1,\ldots,n$. Define

$$C_{r,n}:=\max{\log(r/n),1},\qquad \sigma^2=\bigg\|\sum_{i=1}^n \text{{\bf Var}}[\xi_i]\mathbf{A}_i^2\bigg\|.$$

The rank-sensitive matrix discrepancy conjecture. One has

$${\rm Disc}(\mathbf{A}_1,\ldots,\mathbf{A}_n;\xi_1,\ldots,\xi_n)\leq O(\sqrt{C_{r,n}}\cdot \sigma).$$

The conjecture predicts that the logarithmic factor in matrix discrepancy depends on the matrix rank rather than the ambient dimension; the source presents it as a conjectural extension of the preceding bounds, and no resolution is supplied.