Weak variance conjecture for the spectral norm of Gaussian matrices

Let $X$ be a $d\times d$ centered random matrix with jointly Gaussian entries. Define its weak variance by

$$\sigma_{*}(X)=\sup_{\substack{v,w\in\mathbb{R}^{d}\\\|v\|_{2}=\|w\|_{2}=1}}\left(\mathbb{E}\langle Xv,w\rangle^{2}\right)^{\frac{1}{2}}.$$

Weak variance conjecture. The expected spectral norm satisfies

$$\mathbb{E}\|X\|\lesssim\|\mathbb{E}(X^{*}X)\|^{\frac{1}{2}}+\|\mathbb{E}(XX^{*})\|^{\frac{1}{2}}+\sqrt{\log d}\,\sigma_{*}(X).$$

The conjecture proposes that the weak variance is the parameter governing whether a logarithmic factor is necessary in noncommutative Khintchine-type bounds. The supplied text does not establish the claim or give evidence of its resolution, so its status remains open.