The resonance formula for the linear strand of an arrangement's Orlik-Solomon algebra

Let ${\mathcal A}$ be a complex hyperplane arrangement, let $A=H^*(X({\mathcal A}),\Bbbk)$ be its Orlik-Solomon algebra over a field of characteristic $0$, and let $E$ be the exterior algebra on the degree-one generators of $A$. Write

$$\beta_{i,j}=\dim_{\Bbbk}\operatorname{Tor}^{E}_{i}(A,\Bbbk)_j$$

and let $h_r$ be the number of $r$-dimensional components of the projective first resonance variety ${\mathcal R}^{1}({\mathcal A})$. Resonance formula for the linear strand. For $k\gg 0$, the graded Betti numbers in the linear strand satisfy

$$\beta_{k-1,k}=(k-1)\sum_{r\geq 1}h_r\binom{r+k-1}{k}.$$

This was presented as a pattern suggested by examples; a purely combinatorial formula for these linear-strand Betti numbers was unknown in the stated generality.