Conjecture on norm comparison for matrices with independent columns

Let $A$ be an $m \times n$ matrix whose columns $A_i$ are independent, mean zero, subgaussian random vectors in $\mathbb{R}^m$ with $\norm*{A_i}_{\psi_2} \leq K$. For a subset $T \subset \mathbb{R}^n$, let $\gamma(T)$ denote the relevant geometric complexity functional, and let $D$ be the diagonal matrix satisfying

$$D_{i,i}=\norm{A_i}_2.$$

Norm-comparison conjecture. One has

$$\operatorname{\mathbb{E}}\sup_{x\in T}\abs[\Big]{\norm{Ax}_2-\norm{Dx}_2}\leq C(K)\gamma(T),$$

where $C(K)$ depends only on $K$. This conjecture asks whether the normalized-column assumption can be removed from the paper's norm-comparison result for random matrices with independent columns.