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
