Iyengar–Ma–Walker length–multiplicity conjecture

Let $R$ be a local ring, and let $M$ be a nonzero $R$-module of finite projective dimension. Iyengar–Ma–Walker length–multiplicity conjecture.

$$\ell_R(M)\geq e(R).$$

This is one of two multiplicity conjectures introduced by Iyengar, Ma, and Walker. The source presents it as an open conjecture concerning the relationship between module length and multiplicity.