Asymptotic Lech Bounds and Bertini Flags

For a $d$-dimensional Noetherian local ring $(R,\mathfrak{m})$, with $d>0$, let $c_{\infty}(R)$ be the limit, as $N\to\infty$, of the suprema of $e_R(I)/(d!\ell_R(R/I))$ over ideals $I$ with $\sqrt{I}=\mathfrak{m}$ and $\ell_R(R/I)>N$. We prove that if $R$ is complete, reduced, equidimensional, and