Total rank conjecture for finite-length modules

Let $(R,\mathfrak{m},k)$ be a $d$-dimensional Noetherian local ring, and let $M$ be a finitely generated nonzero $R$-module. Suppose that $M$ has finite length and finite projective dimension. Total rank conjecture. The total sum of the Betti numbers of $M$ over $R$ satisfies

$$\sum_{i\geq 0}\beta_i^{R}(M)\geq 2^d.$$

This is a weaker version of the Buchsbaum–Eisenbud–Horrocks conjecture. The source introduces it as a conjecture motivated by work of Avramov and Buchweitz; no resolution status is given in the supplied text.