Regularization conjecture for random matrix operator norms

Let $A$ be an $n \times n$ random matrix with i.i.d. mean-zero entries satisfying

$$\operatorname{\mathbb{E}} A_{ij}^2=1.$$

For a suitable $L_m$-norm, let $\tilde{A}$ be obtained from $A$ by zeroing out every row and column satisfying

$$\|\operatorname{row}_i(A)\|_m \geq C\operatorname{\mathbb{E}}\|\operatorname{row}_i(A)\|_m \quad\text{or}\quad \|\operatorname{col}_i(A)\|_m \geq C\operatorname{\mathbb{E}}\|\operatorname{col}_i(A)\|_m.$$

Here $m$ is to be specified, with $m=2$ as an example. Regularization conjecture. There are constants $C,C'$ such that, with probability $1-o(1)$, the operator norm of the resulting matrix satisfies

$$\|\tilde{A}\|\leq C'\sqrt{n}.$$

The conjecture seeks to remove the extra $\sqrt{\ln\ln n}$ factor from the optimal $\sqrt{n}$ order in the authors' regularization result, and also to avoid the assumption that the entry distribution is symmetric. The required choice of $L_m$-norm remains unspecified; the statement is presented as a potential improvement rather than an established theorem.