The generalized Lech inequality for products with the maximal ideal

Let $(R,{\mathfrak m})$ be a Noetherian local ring of dimension $d\geq4$, and let $\lambda(R/I)$ denote the length of $R/I$. The product-form generalized Lech inequality. For every ${\mathfrak m}$-primary ideal $I$,

$$\operatorname{e}({\mathfrak m}I)\leq d!\lambda(R/I)\operatorname{e}(R).$$

This is a stronger-looking product-form inequality related to the preceding mixed-multiplicity bound. The source presents it as a conjecture proposed in the cited work; its general validity is not established there.