Resonance formula for Chen groups of arrangements

Let $\mathcal{A}$ be a complex hyperplane arrangement with arrangement group $G=G(\mathcal{A})$. Write the resonance variety as $R_1(\mathcal{A})=\bigcup_i L_i$, where the $L_i$ are linear subspaces, and let $h_r=\#\{L_i\mid \dim L_i=r\}$. For a free group $F_r$, let $\theta_k(F_r)$ denote its Chen ranks, and let $\theta_k(G)$ denote the Chen ranks of $G$. Resonance formula for Chen groups. One has

$$\theta_k(G)=\sum_{r\ge 2}h_r\theta_k(F_r),\qquad\text{for $k\ge 4$.}$$

Moreover, the Chen groups $\operatorname{gr}_k(G/G”)$ are free abelian of rank $\theta_k$ for all $k\ge 1$. This conjecture makes the expected combinatorial dependence of the Chen groups explicit through the resonance components; the equivalent large-$k$ expression is $\theta_k(G)=(k-1)\sum_{r\ge 2}h_r\binom{k+r-2}{k}$.