Does the distribution of the maximum increase when adding independent Gaussian processes?

Question

Let $x(t)$ and $y(t)$ be independent, mean-zero Gaussian processes, indexed over some general metric space $T$. Is it true that $\Pr(\sup_{t \in T} |x(t) + y(t)| > z) \ge \Pr(\sup_{t \in T} |x(t)| > z)$ for all $z > 0$?