The simplicity conjecture for tangent bundles of Fano manifolds

Let $X$ be a Fano manifold with Picard number one. A vector bundle is simple if its only endomorphisms are scalar multiplications. The simplicity conjecture. The tangent bundle $TX$ is simple. This is presented as a weaker replacement for the disproved semistability conjecture. The source proves it for smooth horospherical varieties of Picard number one, while the general case remains open in the supplied text.