The MLS LCS formula for minimal linear strand arrangements

Let ${\mathcal A}$ be a complex hyperplane arrangement with group $G$, and let $L_2({\mathcal A})$ denote the set of rank-two elements in its intersection lattice. For $X\in L_2({\mathcal A})$, let $\mu(X)$ be the Möbius-function value of $X$, and let $F_{\mu(X)}$ be the free group of rank $\mu(X)$. Write $\phi_k(G)$ for the rank of the $k$th lower-central-series quotient of $G$. The arrangement is minimal linear strand (MLS) when its linear strand in the free resolution of its Orlik–Solomon algebra over the exterior algebra is minimal.

MLS LCS formula. If ${\mathcal A}$ is an MLS arrangement with group $G$, then, for all $k\ge 2$,

$$\phi_k(G)=\sum_{X\in L_2({\mathcal A})}\phi_k(F_{\mu(X)}).$$

Equivalently,

$$\prod_{k=1}^{\infty}(1-t^k)^{\phi_k}= (1-t)^{b_1} \prod_{X\in L_2({\mathcal A})} \frac{1- \mu(X)\, t}{1-t}.$$

The formula extends the computed identities for the first four lower-central-series ranks and is presented as the expected general outcome of the spectral-sequence calculation. Its verification is left open; it is described as the simplest case of the more general Resonance LCS Conjecture.