  Recently, Kanemitsu has discovered a counterexample to the long-standing
conjecture that the tangent bundle of a Fano manifold of Picard number one is
(semi)stable. His counterexample is a smooth horospherical variety. There is a
weaker conjecture that the tangent bundle of a Fano manifold of Picard number
one is simple. We prove that this weaker conjecture is valid for smooth
horospherical varieties of Picard number one. Our proof follows from the
existence of an irreducible family of unbendable rational curves whose tangent
vectors span the tangent spaces of the horospherical variety at general points.
