On page 5 of a recent manuscript by Lugosi-Mendelson, a claim equivalent to the following statement is made:

Suppose $Z$ is a centered, $\mathbf{R}^d$-valued random variable with $\mathbf{E} e^{\lambda \langle Z, z \rangle} \leq e^{\lambda^2 \|z\|_2^2/2}$, for all $\lambda \in \mathbf{R}, z \in \mathbf{R}^d$. Let $T = \{x : \|x\|_\ast \leq 1\}$ denote the dual norm ball for $\|\cdot\|$ on $\mathbf{R}^d$ and $Z_t := \langle Z, t \rangle$ for $t \in T$. Then with probability at least $1 - \delta$, $$\sup_{t \in T} Z_t \lesssim w(\Sigma^{1/2} T) + \log^{1/2} (1/\delta) \sup_{t \in T} \|Z_t\|_{L^2},$$ where $\Sigma := \mathbf{E} ZZ^T$ denotes covariance and $w(S) := \mathbf{E}_{g \sim N(0, I_d)} \sup_{s \in S}\langle g, s \rangle$ is the Guassian mean-width for $S \subset \mathbf{R}^d$.

(Above $\lesssim$ hides a universal constant.) They say it is a consequence of majorizing measures and the generic chaining. A naive application would give $\gamma_2(T, \|\cdot\|_2)$ (which could be upper bounded by $w(T)$) and ${\rm diam}(T)$ (with resepct to $\ell_2^d$.

Could anyone either sketch the argument for majorizing measures/generic chaining or point me to a reference for this result?