Han–Li conjecture for the constant scalar curvature and mean curvature problem

Let $(M,g_0)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ with boundary $\partial M$. For constants $c_1,c_2\in\mathbb{R}$, consider the problem

$$\left\{\begin{array}{ll} -\frac{4(n-1)}{n-2}\Delta_{g_0}u+R_{g_0}u=c_1u^{\frac{n+2}{n-2}}&\text{ in }M,\\\\ \frac{2}{n-2}\frac{\partial u}{\partial \nu_{g_0}}+h_{g_0}u=c_2u^{\frac{n}{n-2}}&\text{ on }\partial M, \end{array}\right.$$

where $u$ is required to be positive, and let $Y(M,\partial M)$ denote the positive conformal invariant associated with this problem. Han–Li conjecture. If $Y(M,\partial M)>0$, then the problem is solvable for every positive constant $c_1$ and every $c_2\in\mathbb{R}$. This conjecture concerns existence of conformal metrics with constant scalar curvature and constant boundary mean curvature; the supplied text gives no resolution status.