The minimal length conjecture for finite-projective-dimension modules

Let $(R,\mathfrak m)$ be a local ring, let $e(R)$ denote its Hilbert–Samuel multiplicity, and let $\ell_R M$ denote the length of an $R$-module $M$. Minimal length conjecture. For every nonzero $R$-module $M$ of finite projective dimension,

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

This conjecture concerns the smallest possible length of a nonzero module of finite projective dimension. It implies Lech's conjecture at least for Cohen–Macaulay rings, but remains open in the stated generality.