Asymptotics of supremum distribution of a Gaussian process over a Weibullian time

Let $\{X(t):t\in[0,\infty)\}$ be a centered Gaussian process with stationary increments and variance function $\sigma^2_X(t)$. We study the exact asymptotics of ${\mathbb{P}}(\sup_{t\in[0,T]}X(t)>u)$ as $u\to\infty$, where $T$ is an independent of $\{X(t)\}$ non-negative Weibullian random variable. As an illustration, we work out the asymptotics of the supremum distribution of fractional Laplace motion.