The alternative conjecture on the spectral norm of Gaussian random matrices

Let $X$ be the $d\times d$ symmetric random matrix with entries $X_{ij}=b_{ij}g_{ij}$, where $\{g_{ij}:i\ge j\}$ are independent standard Gaussian random variables and $\{b_{ij}:i\ge j\}$ are given nonnegative scalars. For a symmetric matrix, $X^2$ denotes its square, and $\|\cdot\|$ denotes the operator norm. Alternative spectral-norm conjecture. The expected spectral norm satisfies

$$\mathbf{E}\|X\|\asymp\|\mathbf{E}X^2\|^{1/2}+\mathbf{E}\max_{ij}|X_{ij}|.$$

This formulation is motivated by a dimension-dependent upper bound whose first term equals $\|\mathbf{E}X^2\|^{1/2}$ and whose second term controls the expected largest entry. The source presents it as an alternative to Latała's conjecture and does not provide a resolution.