The CR sphere's k-Yamabe classification conjecture

Let $(\Bbb{S}^{2n+1},\hat{\theta})$ be the CR sphere of dimension $2n+1$, and let $k\geq 1$. A pseudohermitian structure $\widetilde{\theta}=e^{2u}\hat{\theta}\in[\hat{\theta}]_{+}$ has constant pseudohermitian $k$-curvature.

CR sphere k-Yamabe conjecture. If the pseudohermitian structure $\widetilde{\theta}=e^{2u}\hat{\theta}\in[\hat{\theta}]_{+}$ has constant pseudohermitian $k$-curvature, then $\widetilde{\theta}$ is equal to a multiple of the canonical form $\hat{\theta}$ by some CR-automorphism of $\Bbb{S}^{2n+1}$.

This is an Obata-type classification result for the CR $k$-Yamabe problem on the sphere. The statement is presented as an open question extending the known classification in the Cotton-admissible class.