Boij–Söderberg convex-hull conjecture for Betti diagrams

Let $m=(m_0,\ldots,m_s)$ and $M=(M_0,\ldots,M_s)$ be strictly increasing sequences of integers. Let $V_{m,M}$ be the vector space of matrices $(\beta_{i,j})$ satisfying the codimension-$s$ Herzog–Kühl equations and supported in the ranges $m_i\leq j\leq M_i$. Let $B_{m,M}$ be the additively closed subset consisting of Betti diagrams in this space, let $\bar B_{m,M}$ be the set of their normalizations by $\beta_0$, and let $\Pi_{m,M}$ be the set of pure diagrams $\pi(d)$ with $m_i\leq d_i\leq M_i$, where

$$\pi(d)_{i,j}=\begin{cases}(-1)^{i+1}\displaystyle\prod_{\substack{k\neq i\k\neq 0}}\frac{d_k-d_0}{d_k-d_i}&\text{if }j=d_i,\\0&\text{if }j\neq d_i. \end{cases}$$

Boij–Söderberg convex-hull conjecture. The set $\bar B_{m,M}$ is the convex hull of $\Pi_{m,M}$.

This conjecture asserts that normalized Betti diagrams with prescribed bounds on their shifts are exactly the convex combinations of pure diagrams in the same region. The source attributes it to Boij and Söderberg but gives no resolution status.