The conjectured optimal smoothed tail bound for the condition number

Let $\widetilde{A}$ be an $n\times n$ matrix satisfying $\|\widetilde{A}\|_2\leq\sqrt{n}$, and let $A$ be a Gaussian perturbation of $\widetilde{A}$ with variance $\sigma^2\leq1$. Optimal condition-number tail conjecture. The condition number should satisfy

$$\Pr\{\kappa(A)\geq x\}\leq O\left(\frac{n}{x\sigma}\right).$$

This would improve the stated bound of order $O(n\log n/(x\sigma))$ and thereby sharpen the smoothed analysis of condition numbers; the paper presents it as an unresolved conjecture.