Contact Fano homogeneity conjecture

Let $X$ be a projective contact manifold. A contact Fano manifold is a contact manifold that is Fano, and a manifold is homogeneous if a connected algebraic group acts transitively on it.

Contact Fano homogeneity conjecture. A contact Fano manifold is homogeneous.

The source explains that this is equivalent to the classification claim that every projective contact manifold is either a projective cotangent bundle or the unique closed orbit in the projectivization of a simple complex Lie algebra. The non-Fano alternative is known, while the Fano homogeneity assertion is presented as open; a positive answer would have consequences for compact quaternion-Kähler manifolds.