Let $X$ be a normal variety over a field of characteristic zero with rational singularities.

If $\pi:Y \to X$ is a birational proper morphism with $Y$ also normal, then does $Y$ also have rational singularities?

It is easy to see that this is true if $\dim(X) = 2$, but the higher dimensional case seems more difficult and perhaps it is even false. If true, I would also be interested in analogous results in positive or mixed characteristic e.g., for pseudo-rational singularities.