Smoothed-analysis conjecture on random perturbations and condition numbers

Let $A$ be an arbitrary $n \times n$ matrix, and let $M_n$ be a random $n \times n$ matrix. For an invertible matrix $M$, define its condition number by

$$\kappa(M):=\|M\|\,\|M^{-1}\|,$$

and set $\kappa(M)=\infty$ when $M$ is not invertible. Call $M$ well-conditioned when $\kappa(M)\leq n^C$ for some constant $C$ independent of $n$. Random-perturbation condition-number conjecture. With high probability, $A+M_n$ is well-conditioned. This intuition motivates smoothed analysis, though the source notes that well-conditioning alone need not account for every algorithmic difficulty. The source also says that rigorous versions of this intuition were used in the cited analysis; the precise perturbation distribution and meaning of “with high probability” are not specified in the conjectural statement.