The noncommutative Gaussian norm conjecture

Let $X=\sum_{k=1}^s g_kA_k$ be the Gaussian matrix model from the noncommutative Khintchine theorem, where $g_k$ are independent standard Gaussian variables and $A_k$ are the associated matrices. Let $B$ be the unit ball used in that theorem, and write $\|X\|$ for the relevant operator norm. Noncommutative Gaussian norm conjecture.

$$\mathbf{E}\|X\|\lesssim \|\mathbf{E}X^2\|^{1/2}+\sup_{v\in B}\mathbf{E}[\langle v,Xv\rangle^2]^{1/2}\sqrt{\log n}.$$

This is explicitly described as speculative and would extend the nearly optimal independent-entry bounds to the dependent setting of noncommutative Khintchine inequalities. The source emphasizes that independence is crucial in the existing proofs and gives no evidence of a resolution.