The submersion version of generalized Chen's conjecture

Let $M^m(c)$ be an $m$-dimensional space form of constant sectional curvature $c$, let $N^{m-1}$ be a manifold, and let $\varphi:M^m(c)\to N^{m-1}$ be a Riemannian submersion. Submersion version of generalized Chen's conjecture. Are biharmonic submersions $\varphi:M^m(c)\to N^{m-1}$ harmonic? Are triharmonic submersions $\varphi:M^m(c)\to N^{m-1}$ harmonic? The biharmonic case is known when $m=3$ and the target has dimension $2$, while this paper gives an affirmative partial answer for triharmonic Riemannian submersions from three-dimensional space forms into surfaces; the general questions remain open.