General independent-entry matrix norm conjecture

Let $(X_{ij})_{i,j\leq n}$ be independent, mean-zero random variables satisfying $\|X_{ij}\|_{2p}\leq\alpha\|X_{ij}\|_p$ for every $p\geq 1$ and all $i,j\leq n$, with a fixed constant $\alpha\geq 1$ (condition (25) in the source). Write $\|\cdot\|$ for the operator norm and $\|S\|_p=({\mathbb E}|S|^p)^{1/p}$.

General independent-entry matrix norm conjecture. Then

$$\begin{aligned} {\mathbb E}\|(X_{ij})\| \sim_{\alpha} &\max_{i}\left(\sum_{j} {\mathbb E} X_{ij}^2\right)^{1/2}+\max_{j}\left(\sum_{i} {\mathbb E} X_{ij}^2\right)^{1/2} \\ &+\max_{1\leq k\leq n}\min_{I\subset [n],|I|\leq k} \sup_{\|s\|_2,\|t\|_2\leq 1}\left\|\sum_{i,j\notin I}X_{ij}s_it_j\right\|_{\log (k+1)}. \end{aligned}$$

This is stated as a generalization of the Rademacher conjecture and would give a matching, up to constants depending on $\alpha$, upper bound for the corresponding lower estimate.