The smallest singular value conjecture for Gaussian matrix perturbations

Let $\widetilde{A}$ be an arbitrary square matrix in $\mathbb{R}^{n\times n}$, and let $A$ be a Gaussian perturbation of $\widetilde{A}$ with variance $\sigma^2$. Smallest singular value conjecture. It should satisfy

$$\Pr\left\{\|A^{-1}\|_2\geq x\right\}\leq\frac{\sqrt{n}}{x\sigma}.$$

This would extend Edelman's sharp Gaussian-matrix bound from perturbations of the zero matrix to arbitrary centers; the paper proves the same estimate with constant $2.35$ instead of $1$.