Triharmonic Riemannian submersions from 3-dimensional manifolds of   constant curvature
  For biharmonic maps, there is a famous conjecture named Chen's conjecture. In later paper, Wang and Ou gave an affirmative partial answer to submersion version of Chen's conjecture. In this paper, we give an affirmative partial answer to submersion version of generalized Chen's conjecture, that is, triharmonic Riemannian submersions from a 3-dimensional space form into a surface is harmonic. 
Tomoya Miura and Shun Maeta
math.DG