The CR k-Yamabe existence conjecture below the spherical constant

Let $(M,\theta)$ be a compact CR manifold of dimension $2n+1$, and let $k\geq 1$. Denote by $\lambda_k^{+}(M)$ the CR $k$-Yamabe constant and by $\lambda_k^{+}(\Bbb{S}^{2n+1})$ its value on the CR sphere.

CR $k$-Yamabe existence conjecture. If

$$\lambda_k^{+}(M)<\lambda_k^{+}(\Bbb{S}^{2n+1}),$$

then the CR $k$-Yamabe constant $\lambda_k^{+}(M)$ is attained by a positive smooth function $u$. In particular, the CR $k$-Yamabe problem has a solution provided that it is variational and

$$\lambda_k^{+}(M)<\lambda_k^{+}(\Bbb{S}^{2n+1}).$$

This conjecture seeks an existence theory below the spherical threshold, analogous to results for the Riemannian and CR Yamabe problems and to fully nonlinear Hessian equations.