Weak Hanes conjecture for boundedly generated ideals

Let $d\geq 2$ and $N\geq d$ be positive integers. Let $(R,\operatorname{\mathfrak{m}})$ be a Noetherian local ring of dimension $d$, and let $I$ be an $\operatorname{\mathfrak{m}}$-primary ideal that can be generated by $N$ elements. Write $\operatorname{e}(I)$ for the Hilbert–Samuel multiplicity of $I$, $\operatorname{e}(R)=\operatorname{e}(\operatorname{\mathfrak{m}})$, and $\ell(R/I)$ for the length of $R/I$. Weak Hanes conjecture. There exists a constant $c=c(N,d)\in(0,1)$ such that

$$\operatorname{e}(I)\leq d!c\operatorname{e}(R)\ell(R/I).$$

This is the expected extension of the proved weak Hanes corollary from integrally closed ideals to arbitrary $\operatorname{\mathfrak{m}}$-primary ideals. The supplied text says it is known in equal characteristic but does not establish it in general.