Boij–Söderberg decomposition conjecture for Cohen–Macaulay Betti diagrams

Let $R$ be the polynomial ring under consideration, and let a Betti diagram record the graded Betti numbers of an $R$-module. A pure diagram is a diagram whose maximal and minimal shifts are both a strictly increasing sequence of integers.

Boij–Söderberg conjecture. The Betti diagram of any Cohen–Macaulay $R$-module is a non-negative linear combination of pure diagrams, and every pure diagram is a rational multiple of the Betti diagram of some Cohen–Macaulay $R$-module.

This conjecture describes the cone of Betti diagrams of Cohen–Macaulay modules in terms of pure diagrams and also predicts that every pure diagram is realized, up to rational scaling, by a Cohen–Macaulay module. The source gives no resolution status.