Let $X$ be a variety of general type.

Assume that $\dim X = 3$. In https://eudml.org/doc/164223 it is proven that $X$ has only finitely many minimal models (i.e., only $\mathbb Q$-factorial terminal singularities and nef canonical bundle) is finite.

Is this statement now also known when $\dim X > 3$?

More precisely, does $X$ only have finitely many minimal models? Is the set of extremal rays on $X$ known to be finite?

Note that we do not need to know whether minimal models exist for this question to make sense. Nevertheless, this is now known by http://arxiv.org/abs/math/0610203 . A positive answer to this question seems to be suggested by "Flops connect minimal models" (Kawamata) . Do I understand correctly?