Lee's global pseudo-Einstein structure conjecture

A strictly pseudoconvex CR $(2n+1)$-manifold is pseudo-Einstein when its pseudohermitian Ricci curvature tensor is function-proportional to its Levi metric; for $n=1$, this condition is equivalently expressed by $W_\alpha=0$. Let $c_1(T_{1,0}M)$ denote the first Chern class of the CR holomorphic tangent bundle.

Lee's conjecture. Any closed strictly pseudoconvex CR $(2n+1)$-manifold with vanishing first Chern class

$$c_1(T_{1,0}M)=0$$

admits a global pseudo-Einstein structure.

The vanishing of the first Chern class is the obstruction identified for the existence of pseudo-Einstein contact forms, and a pseudo-Einstein contact form has vanishing CR $Q$-curvature. The source does not state whether this conjecture has been resolved.