The conjecture that Fano contact manifolds with cyclic Picard group are homogeneous

Let $X$ be a Fano manifold with a contact structure $D \subset T_X$, and let $L=T_X/D$ be the quotient line bundle. Assume that $\operatorname{Pic} X \cong \mathbb{Z}\cdot L$. Fano contact homogeneity conjecture. Then $X$ is homogeneous. This is a well-known conjecture in complex contact geometry, predicting that Fano contact manifolds with Picard group generated by the contact line bundle are homogeneous. The source provides no resolution status or further partial results, so the conjecture is treated as open.