Supremum of infinite-horizon gaussian process that converges to 0 at infinity

Suppose $$X(t),t\in[0,\infty)$$ is a centered gaussian process with covariance function $$\Gamma(t,s)$$, such that $$\Gamma(t,t)$$ is uniformly bounded over $$t\in[0,\infty)$$, and $$\Gamma(t,t)\rightarrow 0$$ as $$t\rightarrow\infty$$. Is it true that $$\sup_{t\in[0,\infty)} |X(t)|$$ is $$O_p(1)$$? Or is some decay rate for the variance required for the supremum to be $$O_p(1)$$?