The biconservative hypersurfaces of Euclidean spaces have conservative stress-energy with respect to the bienergy functional. We study Lorentzian hypersurfaces of Minkowski spaces, satisfying an extended condition (namely, $L_1$-biconservativity condition), where $L_1$ (as an extension of the Laplace operator $\Delta=L_0$) is the $\textit{linearized operator}$ arisen from the first normal variation of $2$nd mean curvature vector field. A Lorentzian hypersurface $\x{}: M_1^n\rightarrowŁ^{n+1}$ is said to be $L_1$-biconservative if the tangent component of vector field $L_1^2x$ is identically zero. The geometric motivation of this subject is a well-known conjecture of Bang-Yen Chen saying that the only biharmonic submanifolds (i.e., satisfying condition $L_0^2 x=0$) of Euclidean spaces are the minimal ones. We discuss on $L_1$-biconservative Lorentzian hypersurfaces of the Lorentz-Minkowski space $Ł^{n+1}$. After illustrating some examples, we prove that these hypersurfaces, with at most two distinct principal curvatures and constant ordinary mean curvature, have constant $2$nd mean curvature.