Descent of nearby cycle formula for Newton non-degenerate functions

We prove a descent theorem of nearby cycle formula for Newton non-degenerate functions at the origin (as well as its motivic version) without assuming the convenience condition. In the isolated singularity case, these imply some well-known formula for the number of Jordan blocks of the Milnor monodromy with the theoretically maximal size, using a standard estimate of weights (where we do not need a theory of the duals of logarithmic complexes on toric varieties). They also provide a proof of a modified version of Steenbrink conjecture on spectral pairs for non-degenerate functions with simplicial Newton polyhedra in the isolated singularity case (which is false in the non-simplicial case).