We prove a relative statement about the slices of the algebraic cobordism spectrum. If the map from MGL to a certain quotient of MGL introduced by Hopkins and Morel is the map to the zero-slice then a relative version of Voevodsky's conjecture on the slices of MGL holds true. We outline the picture for K-theory and rational slices.
"Relations between slices and quotients of the algebraic cobordism spectrum." Homology Homotopy Appl. 12 (2) 335 - 351, 2010.