The linear-strand Betti-number conjecture for pure ideals

Let $E$ be an exterior algebra and let $J\subset E$ be a pure ideal, meaning an ideal generated by products of linear forms. Suppose that $E/J$ satisfies property (*): any two distinct irreducible components of $R^1(E/J)$ meet only at $0$, and if $u,v$ belong to the same irreducible component of $R^1(E/J)$, then $u\wedge v\in J$. For $i\gg 0$, let $h_r$ denote the number of $r$-dimensional components of $R^1(E/J)$ in the affine space $K^n$. The linear-strand Betti-number conjecture. The graded Betti numbers of the linear strand of $E/J$ satisfy

$$\beta^E_{i,i+1}(E/J)= i\sum_{r\geq 1} h_r \binom{r+i-1}{i+1}.$$

This conjecture generalizes a conjecture of Schenck and Suciu for Orlik–Solomon algebras. It is disproved for arbitrary pure ideals, although additional hypotheses may still yield such a formula.