Avramov–Buchweitz total rank conjecture

Let $R$ be a $d$-dimensional Noetherian local ring, and let $M$ be a finitely generated nonzero $R$-module. Total rank conjecture. If $M$ has finite length and finite projective dimension, then

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

This is described as a weaker version of the Buchsbaum–Eisenbud–Horrocks conjecture. It has been proved in several cases, including certain complete intersections, but remains open in general.