A submanifold Mn of the Euclidean space En+m is said to be biharmonic if its position map x : M^n → E^{n+m} satisfies the condition ∆^2 x = 0, where ∆ stands for the Laplace operator. A well-known conjecture of Bang-Yen Chen says that, the only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider a modified version of the conjecture, replacing ∆ by its extension, L_1-operator (namely, L_1-conjecture). The L_1-conjecture states that any L_1-biharmonic Euclidean hypersurface is 1-minimal. We prove that the L1-conjecture is true for L_1-biharmonic hypersurfaces with three distinct principal curvatures and constant mean curvature of a Euclidean space of arbitrary dimension.
