  A $k$-harmonic map is a critical point of the $k$-energy in the space of
smooth maps between two Riemannian manifolds. In this paper, we prove that if
$M^n (n\ge 3)$ is a CMC proper triharmonic hypersurface with at most three
distinct principal curvatures in a space form $\mathbb{R}^{n+1}(c)$, then $M$
has constant scalar curvature. This supports the generalized Chen's conjecture
when $c\le 0$. When $c=1$, we give an optimal upper bound of the mean curvature
$H$ for a non-totally umbilical proper CMC $k$-harmonic hypersurface with
constant scalar curvature in a sphere. We also prove that a proper CMC
triharmonic with two distinct principal curvatures in a sphere must be locally
a generalized Clifford torus with suitable parameters. At last, we give the
complete classification of the 3-dimensional closed proper CMC triharmonic
hypersurfaces in $\mathbb{S}^{4}$.
