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The Han-Li conjecture states that: Let $(M,g_0)$ be an $n$-dimensional$(n\geq 3)$ smooth compact Riemannian manifold with boundary having positive(generalized) Yamabe constant and $c$ be any real number, then there exists aconformal metric of $g_0$ with scalar curvature $1$ and boundary mean curvature$c$. Combining with Z. C. Han and Y. Y. Li's results, we answer this conjectureaffirmatively except for the case that $n\geq 8$, the boundary is umbilic, theWeyl tensor of $M$ vanishes on the boundary and has a non-zero interior point.

The Han-Li conjecture in constant scalar curvature and constant boundary mean curvature problem on compact manifolds