Non-tensorial norm estimate for random matrices with Weibull entries

Assume that $r\range[1,2]$, $(a_{i,j})_{i\le m,j\le n}$ is a deterministic $m\times n$ matrix, and $(X_{i,j})_{i\le m,j\le n}$ are independent identically distributed symmetric Weibull random variables with parameter $r$. Let

$$D_{3,r}=\begin{cases} {\mathbb E}\max_{i\le m,j\le n}|a_{i,j}X_{i,j}| & \text{if }p\le 2\le q,\\ \max_{j\le n}b_j^*\ln^{1/r}(j+1) & \text{if }p\le q\le 2,\\ \max_{i\le m}d_i^*\ln^{1/r}(i+1) & \text{if }2\le p\le q,\\ 0 & \operatorname{if }q<p. \end{cases}$$

Is it true that

$${\mathbb E}\left\|(a_{i,j}X_{i,j})_{i,j}:\ell_p^n\to\ell_q^m\right\|\sim_{p,q}D_1+D_2+D_{3,r}?$$

Non-tensorial Weibull norm conjecture. The expected operator norm is comparable, with constants depending on $p$ and $q$, to $D_1+D_2+D_{3,r}$.