  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. As an application, we give the complete
classification of the 3-dimensional closed proper CMC triharmonic hypersurfaces
in $\mathbb{S}^{4}$.
