Uniform Lech's inequality

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d\geq 2$. We prove that if $e(\widehat{R}_{red})>1$, then the classical Lech's inequality can be improved uniformly for all $\mathfrak{m}$-primary ideals, that is, there exists $\varepsilon>0$ such that $e(I)\leq d!(e(R)-\varepsilon)\ell(R/I)$ for all $\mathfrak{m}$-primary ideals $I\subseteq R$. We also obtain partial results towards improvements of Lech's inequality when we fix the number of generators of $I$.