Existence conjecture for negative scalar curvature metrics on manifolds of positive type

Let $(M,g)$ be a compact Riemannian manifold with boundary $\partial M$, of positive type, and dimension $n\geq 3$. Given $K<0$ and $c>\sqrt{-(n-2)K/n}$, a smooth solution $u>0$ to the conformal boundary-value equations exists.

Existence conjecture. Given $K<0$ and $c>\sqrt{-(n-2)K/n}$, there exists a smooth solution $u>0$ to equations.

The conjecture concerns existence in the negative scalar-curvature case on manifolds of positive type. Unlike the cases $K\geq 0$, few existence results are known when $K<0$; the paper presents this as an open problem motivated by model solutions on geodesic balls in hyperbolic space.