The projective-space conjecture for smooth Fano varieties

Let $X$ be a smooth Fano variety of Picard number one, meaning that $-K_X$ is ample and $\rho(X)=1$. Suppose that $X$ admits a non-isomorphic surjective endomorphism $f$. The projective-space conjecture. Then $X$ is a projective space. This is a long-standing conjecture from the 1980s; it is known for homogeneous spaces, Fano threefolds of Picard number one, and hypersurfaces of projective space, but remains open in general.