The folklore conjecture on endomorphisms of Fano manifolds of Picard number 1

Let $X$ be a Fano manifold of Picard number 1, and let a non-isomorphic surjective endomorphism of $X$ mean a surjective morphism $f\colon X\to X$ that is not an isomorphism.

Folklore conjecture. If $X$ admits a non-isomorphic surjective endomorphism, then $X$ is a projective space.

This folklore conjecture from the 1980s concerns the classification of Fano manifolds of Picard number 1 with non-trivial self-maps. The paper proves dynamical rigidity under additional hypotheses, including bigness of the tangent bundle and a condition on the variety of minimal rational tangents, but the conjecture as stated remains open.