In this paper, we introduce $L_k$-biharmonic hypersurfaces $M$ in simply connected space forms $R^{n+1}(c)$ and propose $L_k$-conjecture for them. For $c = 0,−1$, we prove the conjecture when hypersurface $M$ has two principal curvatures with multiplicities $1,n−1$, or $M$ is weakly convex, or $M$ is complete with some constraints on it and on $L_k$. We also show that neither there is any $L_k$-biharmonic hypersurface $M^n$ in $\mathbb {H}^{n+1}$ with two principal curvatures of multiplicities greater than one, nor any $L_k$-biharmonic compact hypersurface $M^n$ in $\mathbb {R}^{n+1}$ or in $\mathbb {H}^{n+1}$. As a by-product, we get two useful, important variational formulas. The paper is a sequel to our previous paper, (Taiwan. J. Math., 19, 861–874, 5) in this context.