The conformal-bienergy functional \(E_2^c\) is a modified version of the classical bienergy functional \(E_2\) and it is conformally invariant in the case of a four-dimensional domain. The critical points of \(E_2^c\) are called conformal-biharmonic and denoted c-biharmonic. In the first part of the paper we study the c-biharmonic hypersurfaces \(M^m\) with constant principal curvatures in the product space \({\mathbb {L}}^m(\varepsilon ) \times \mathbb {R}\) , where \({\mathbb {L}}^m(\varepsilon )\) denotes a space form of constant sectional curvature \(\varepsilon \) . Specifically, we demonstrate that \(M^m\) is either totally geodesic or a cylindrical hypersurface of the form \(M^{m-1} \times \mathbb {R}\) , where \(M^{m-1}\) is an isoparametric c-biharmonic hypersurface in \({\mathbb {L}}^m(\varepsilon )\) . In the second part of this article we obtain a full description of isoparametric c-biharmonic hypersurfaces in \({\mathbb {S}}^{m+1}\) and a complete classification of c-biharmonic hypersurfaces with constant scalar curvature in \({\mathbb {S}}^{m+1}\) , \(m=2,3\) and \(m=4\) with an additional assumption. In this context, we shall also prove a global result for compact c-biharmonic hypersurfaces in \({\mathbb {S}}^5\) . In the final part of the paper, as a preliminary effort to understand c-biharmonic hypersurfaces in \({\mathbb {L}}^m(\varepsilon ) \times \mathbb {R}\) with non-constant mean curvature, we establish that a totally umbilical c-biharmonic hypersurface must necessarily be totally geodesic.
