It is a known fact (Borell-TIS inequality) that, given an almost surely bounded Gaussian centered process $X(t), t \in T$, where $T$ is a topological space, $$\mathbb{P}\{\sup_t X(t)-\mathbb{E} \sup_t X(t)>u\}\leq \exp\left(-\frac{u^2}{2\sigma_T^2}\right)\},$$ where $\sigma_T^2=\sup_{t \in T} \mathbb{E} X(t)^2$ for $u>0$. In particular, $\sigma_T$ and $\mathbb{E} \sup_t X(t)$ are finite.

Is it true that if instead $X(t), t \in T$ is not necessarily Gaussian and not necessarily centered, but almost surely bounded from above and subgaussian in the sense that $$\limsup_{u \to \infty} \sup_{t \in T} \frac{\log\mathbb{P}\{X(t)>u\}}{u^2}\leq -\frac{1}{2\sigma^2}$$ for some $\sigma>0$, then also $$\limsup_{u \to \infty}  \frac{\log\mathbb{P}\{\sup_{t \in T} 
 X(t)>u\}}{u^2}\leq -\frac{1}{2\sigma^2}?$$