Expected value of the kth largest of n not necessarily independent Gaussians

Question

Let $X = (X_1, X_2, \ldots, X_n)$ be a Gaussian vector where each $X_i$ is distributed as $\mathcal{N}(0, 1)$. Then,

$$\mathbb{E}\left[\max_{i \leq n} X_i\right] \leq C\sqrt{\log n}$$

for some constant $C$ regardless of whether or not the $X_i$ are independent.

Let $f_r(X)$ be the $r$th largest coordinate of $X$. Then assuming that the $X_i$ are independent, $$\mathbb{E}[f_r(X)] \leq C\sqrt{\log n/r},$$ for some constant $C$. Does this still hold if the $X_i$ are not independent?