Question about Ornstein–Uhlenbeck process

Question

Suppose $X(t) = e^{-t/2}W(e^{t})$ is an Ornstein–Uhlenbeck process ($W(t)$ is a Wiener process) and $\epsilon > 0$. I'm trying to find a constant $\delta>0$ so that $$ \lim_T\mathbb{P}\Big( \sup_{0 \leq t\leq T}|X(t)|^2 + \epsilon < \sup_{0 \leq t\leq T(1+\delta)}|X(t)|^2\Big) \approx 1. $$ By a version of the law of the iterated logarithm, I know that $$ \lim_T (2\log(T))^{-1}\sup_{0 \leq t \leq T}|X(t)|^2 = 1 \quad a.s., $$ so that $$ \sup_{0 \leq t \leq T}|X(t)|^2 \approx 2\log(T), \quad \sup_{0 \leq t \leq T(1+\delta)}|X(t)|^2 \approx 2\log(T(1+\delta)) = 2\log(T) + 2\log(1+\delta) \quad a.s. $$ Thus, I'm tempted to just choose $\delta$ so that $ 2\log(1+\delta) > \epsilon $, but I'm worried this intuition is flawed. Any help making this more rigorous would be much appreciated.