Strongly Lech-independent ideals and Lech's conjecture

We introduce the notion of strongly Lech-independent ideals as a generalization of Lech-independent ideals defined by Lech and Hanes, and use this notion to derive inequalities on multiplicities of ideals. In particular we prove that if $(R,\mathfrak{m}) \to (S,\mathfrak{n})$ is a flat local extension of local rings with $\dim R = \dim S$, the completion of $S$ is the completion of a standard graded ring over a field $k$ with respect to the homogeneous maximal ideal, and the completion of $\mathfrak{m}S$ is the completion of a homogeneous ideal, then $e(R) \leq e(S)$.