Compactness and existence conjecture for conformally invariant curvature equations

Let $(N,g)$ be a Riemannian manifold, let $[g]$ denote its conformal class, and let $A_g$ be the curvature tensor whose eigenvalues define the symmetric function $c3_k$. Write $A_g\in\Gamma_k^+$ when its eigenvalues lie in the positive $k$-cone. A conformal deformation has the form

$$\tilde{g}=e^{2u}g.$$

Compactness and existence conjecture. If $A_g\in\Gamma_k^+$, then there exists a conformal deformation $\tilde{g}=e^{2u}g$ such that

$$\sigma_k(A_{\tilde{g}})=1.$$

Furthermore, if $(N,g)$ is not conformally equivalent to $S^n$ with the standard metric, then the space of solutions is compact.

The claim generalizes the Yamabe problem to the fully nonlinear curvature equations associated with the symmetric functions $\sigma_k$. The preceding determinant case gives an existence and compactness result under the stronger condition $\sigma([g])<\frac{\pi^2}{2}$, while the assertion for all $k$ and all nonspherical conformal classes is presented as a conjectural extension.