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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?

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By

C. Hacon, J. McKernan, "On the existence of flips", math.AG/0507597.

finiteness of minimal models in dimension $n-1$ implies the existence of flips in dimension $n$.

I guess the result you are looking for is Theorem B (pag 11) of BCHM (http://www2.imperial.ac.uk/~pcascini/Papers/0610203.pdf).

Perhaps Section 5.5 (pag 20) of this

O. Fujino, "Recent developments in minimal model theory", https://www.math.kyoto-u.ac.jp/~fujino/recent-final.pdf

is easier to read.

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