According to a well-known conjecture of Bang-Yen Chen on
Euclidean spaces, every submanifold with harmonic mean curvature vector field is minimal. Inspired by the conjecture, we study
the Lorentz hypersurfaces of the Minkowski 5-space. The second mean curvature vector field of such a hypersurface is called
harmonic if it is a null vector of the Cheng-Yau operator. we
prove that a hypersurface with harmonic second mean curvature
vector field and three distinct principal curvature is 1-minimal.
We consider different cases based on four possible matrix forms
of the shape operator of Lorentz hypersurface in Minkowski 5-
space