Sasakian uniformization conjecture for positive transverse holomorphic bisectional curvature

Let $M$ be a complete noncompact Sasakian $(2n+1)$-manifold, and let $\mathbb{H}_{n}$ denote the standard Heisenberg group

$$\mathbb{H}_{n}=\mathbb{C}^{n}\times\mathbb{R}.$$

Assume that $M$ has positive transverse holomorphic bisectional curvature.

Sasakian uniformization conjecture. Then $M$ is CR biholomorphic to the standard Heisenberg group $\mathbb{H}_{n}$.

This is proposed as the Sasakian analogue of Yau's uniformization conjecture. The source reports a five-dimensional maximal-volume-growth result, but does not state that the conjecture is proved in its full generality.