Szarek-type singular-value bound for random matrices satisfying the general assumption

Let $\boldsymbol{M}_n$ be a real random matrix satisfying the paper's Assumption with parameter $K>0$, possibly with some moment assumptions on its entries. For a deterministic matrix $A\in\mathbb{R}^{n\times n}$, let $\sigma_{n-k+1}(A+\boldsymbol{M}_n)$ denote the $(n-k+1)$-st singular value. Generalized Szarek conjecture. There is a universal constant $C$ such that, for every such $A$ and every relevant $k$ and $\varepsilon$,

$$\mathbb{P}\left[\sigma_{n-k+1}(A+\boldsymbol{M}_n)\leq \frac{k\varepsilon}{n}\right]\leq (CK\varepsilon)^{k^2}.$$

The claim would extend Szarek's singular-value bound for real Ginibre matrices to the more general class specified by the assumption and would improve the stated general theorem in its dependence on $k$ and $n$. The supplied status evidence says that the claim was proven by Tikhomirov under weaker independence assumptions.