Chen's biharmonic conjecture is well-known and stays open: The only
biharmonic submanifolds of Euclidean spaces are the minimal ones. In
this paper, we consider an advanced version of the conjecture,
replacing Delta by its extension, L_1-operator
(L_1-conjecture). The L_1-conjecture states that any
L_1-biharmonic Euclidean hypersurface is 1-minimal. We prove that
the L_1-conjecture is true for L_1-biharmonic hypersurfaces with
three distinct principal curvatures and constant mean curvature of a
Euclidean space of arbitrary dimension.