First-moment lower-bound conjecture for average-case matrix discrepancy

Let $\bm A_1,\dots,\bm A_T\in\mathbb{R}^{n\times n}_{\operatorname{sym}}$ be i.i.d. random symmetric matrices, where $T=T(n)$, $\|\bm A_i\|\leq1$ almost surely, the distribution is “sufficiently nice,” and the effective rank is $r=r(n)$. Informal discrepancy lower bound. With high probability,

$$\Delta(\bm A_1,\dots,\bm A_T)\gtrsim\sqrt{\frac{rT}{n}}\,4^{-T/n^2}.$$

The phrases “sufficiently nice” distribution and “effective rank” are deliberately left undefined in the source, so this is an informal conjecture about first-moment lower bounds for average-case matrix discrepancy. Its status is open.