Let $(X, \Delta)$ be a (projective) klt pair (say over $\mathbb{C}$, but I am also interested in fields of positive characteristic) and $f: X \to Z$ the contraction associated to a $(K_X + \Delta)$-negative extremal face $F$ of $\overline{NE}(X)$.

Assuming $f$ is a birational morphism, does $Z$ have rational singularities?

If $f$ can be factored as a sequence of divisorial contractions then this is well-known, but I do not know whether this is true if $f$ is a small contraction.

I do not even know if $Z$ is always Cohen-Macaulay (which would be implied by $Z$ having rational singularities).

(As for motivation, the question probably has little or no direct relevance to the MMP; I am interested in rational singularities for their own sake.)