Avramov's total rank conjecture

Let $R$ be a commutative Noetherian ring with connected spectrum, and let $M$ be a nonzero $R$-module of finite projective dimension. Choose a projective resolution

$$0 \to P_d \to \cdots \to P_1 \to P_0 \to M \to 0.$$

Avramov's total rank conjecture. Under the same assumptions as in the Buchsbaum–Eisenbud–Horrocks conjecture,

$$\sum_{i=0}^d \operatorname{rank}_R(P_i)\geq 2^{\operatorname{codim}_R(M)}.$$

This is a weaker sum-of-Betti-numbers form of the Buchsbaum–Eisenbud–Horrocks conjecture. It was proved in the paper for locally complete intersections with $2$-torsion-free modules and for rings of odd characteristic, while the paper proves it in characteristic two.