A well-known conjecture of Bang-Yen Chen says that the only biharmonic Euclidean submanifolds are minimal ones, which affirmed by himself for surfaces in 3-dimensional Euclidean space, E3. We consider an extended version of Chen conjecture (namely, L_k-conjecture) on Lorentzian hypersurfaces of the pseudo-Euclidean space E^4 1 (i.e. the Einstein space). The biconservative submanifolds in the Euclidean spaces are submanifolds with conservative stress-energy with respect to the bienergy functional. In this paper, we consider an extended condition (namely, Lk-biconservativity) on non-degenerate timelike hypersurfaces of the Einstein space E^4_1 . A Lorentzian hypersurface x : M^3_1 → E^4_1 is called L_k-biconservative if the tangent part of (L_k)^2 x vanishes identically. We show that Lk-biconservativity of a timelike hypersurface of E^4_1 (with constant kth mean curvature and some additional conditions) implies that its (k + 1)th mean curvature is constant.
