CR Obata-type conjecture for pseudohermitian manifolds

Let $(M,\theta)$ be a closed pseudohermitian $(2n+1)$-manifold with $n\ge 2$. Assume that the 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

$$$$

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-type conjecture. Under these hypotheses, $(M,\theta)$ must be the standard Sasakian CR structure on the unit sphere in $\mathbb{C}^{n+1}$. This conjecture is an equality-case analogue of the CR Lichnerowicz theorem; the supplied text states that it was established in the non-Sasakian case under additional assumptions on the pseudohermitian torsion, so the full formulation is resolved in the cited work.