LeBrun–Salamon conjecture for Fano contact manifolds

Let $Z$ be a Fano contact manifold, meaning a compact complex contact manifold whose first Chern class can be represented by a positive $(1,1)$-form, and suppose that $b_{1}(Z)=1$.

LeBrun–Salamon conjecture. The manifold $Z$ must be homogeneous.

For Fano contact manifolds with $b_{2}(Z)>1$, the corresponding classification is known: $Z$ is the projectivized cotangent bundle of complex projective space and is homogeneous. The case $b_{1}(Z)=1$ remains open in the stated generality.