Sankar–Teng–Spielman least singular value conjecture for subgaussian random matrices

Let $\xi$ be a mean-zero, variance-at-least-$1$, subgaussian random variable, and let $N_n$ be an $n\times n$ random matrix with iid entries distributed as $\xi$. The least singular value of $N_n$ is denoted by $s_n(N_n)$. Sankar–Teng–Spielman conjecture. There exist constants $C,c>0$ such that, for every $\varepsilon\geq 0$,

$${\mathbb{P}}\bigl(s_n(N_n)\leq \varepsilon n^{-1/2}\bigr)\leq C\varepsilon+c^n.$$

This conjecture seeks a uniform small-ball estimate for the least singular value beyond the Gaussian and smoothed-analysis settings. It would establish the expected $n^{-1/2}$ scale for arbitrary mean-zero, variance-at-least-$1$ subgaussian entries, up to an exponentially small error term; the supplied source gives no resolution status.