Resonance formula for lower central series ranks of arrangements

Let $\mathcal{A}$ be a complex hyperplane arrangement with arrangement group $G=G(\mathcal{A})$. Let $R_1(\mathcal{A})=\bigcup_i L_i$ be its resonance variety, and set $h_r=\#\{L_i\mid\dim L_i=r\}$. Denote by $\phi_k(G)$ the ranks of the lower central series quotients of $G$, and by $\phi_k(F_r)$ those of the free group $F_r$; denote the Chen rank in degree $4$ by $\theta_4(G)$. Resonance LCS formula. If $\phi_4(G)=\theta_4(G)$, then

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

Moreover, the lower central series quotients $\operatorname{gr}_k G$ are free abelian of rank $\phi_k$ for all $k\ge 1$. The proposed formula parallels the lower central series formula for fiber-type arrangements, replacing the exponents by the dimensions of resonance components; its resolution is not supplied in the source.