Sato's conjecture for smooth Fano varieties of Picard number one

Let $X$ be a smooth Fano variety of Picard number one, and suppose that $X$ admits a non-isomorphic surjective endomorphism $f$.

Sato's conjecture. $X$ is a projective space.

This is a long-standing conjecture from the 1980s and is presented as a special case or extension of the rationally connected variety question. The supplied text does not state that it has been resolved in general.