CR Obata conjecture for closed pseudohermitian manifolds

Let $(M,\theta)$ be a closed pseudohermitian $(2n+1)$-manifold with $n\geq 2$. Assume, in addition, that the CR-Paneitz operator is nonnegative if $n=1$. Suppose there is a positive constant $k_0$ such that the pseudohermitian Ricci curvature $Ric$ and pseudohermitian torsion $A$ satisfy the inequality

$$\text{the inequality }.$$

If $\frac{n}{n+1}k_0$ is an eigenvalue of the sub-Laplacian, then $(M,\theta)$ is the standard Sasakian CR structure on the unit sphere in $\mathbb{C}^{n+1}$. CR Obata conjecture. This is the CR analogue of Obata's theorem, characterizing the equality case in a sharp sub-Laplacian eigenvalue estimate. The non-Sasakian case was established under additional assumptions on the divergence and second covariant derivative of the pseudohermitian torsion, while the full statement is resolved according to the supplied status evidence.