The projective-space conjecture for Fano manifolds

Let $X$ be a Fano manifold of Picard number $1$, meaning that $-K_X$ is ample and the Picard number of $X$ is $1$. Projective-space conjecture. If $X$ admits a non-isomorphic surjective endomorphism, then $X$ is a projective space. This long-standing conjecture is known in several classes, including almost homogeneous spaces, Fano threefolds of Picard number $1$, certain Fano manifolds containing a rational curve with trivial normal bundle, and smooth hypersurfaces of projective space; it remains open in general.