Stronger Buchsbaum–Eisenbud–Horrocks conjecture for syzygy ranks

Let $R=k[x_1,\ldots,x_n]$, where $k$ is a field, and let $M$ be a nonzero finitely generated graded $R$-module of codimension $c$. For $i\geq 1$, let $\Omega_i(M)$ denote the $i$th syzygy module of $M$.

Stronger Buchsbaum–Eisenbud–Horrocks conjecture for syzygy ranks.

$$\operatorname{rk}\Omega_i(M)\geq {c-1\choose i-1}.$$

This is the stronger rank formulation proposed by Buchsbaum–Eisenbud and Horrocks; the Betti-number conjecture follows from the rank identity relating consecutive syzygies. Its general status is open.