  A triharmonic map is a critical point of the tri-energy in the space of
smooth maps between two Riemannian manifolds. In this paper, we prove that if
$M^n (n\ge 4)$ is a CMC proper triharmonic hypersurface in a space form
$\mathbb{R}^{n+1}(c)$ with four distinct principal curvatures and the
multiplicity of the zero principal curvature is at most one, then $M$ has
constant scalar curvature. In particular, we obtain any CMC proper triharmonic
hypersurface in $\mathbb{R}^5(c)$ is minimal when $c\le 0$, which supports the
generalized Chen's conjecture. We also give some characterizations of CMC
proper triharmonic hypersurfaces in $\mathbb{S}^5$.
