The cut-number equality conjecture

Let $\Gamma$ be a group. For each $k$, let $c_k(\Gamma)$ denote the largest cardinality of a primitive set in $H^1(\Gamma,\mathbb Z)$ for which all $k$-fold Massey products vanish. Let $F_n$ be the free group of rank $n$, with lower central series

$$F_{n,0}=F_n,\qquad F_{n,k+1}=[F_{n,k},F_n],$$

and let $d_k(\Gamma)$ be the largest $n$ for which there is an epimorphism $\Gamma\to F_n/F_{n,k}$, with $d_\infty(\Gamma)=d(\Gamma)$. The cut-number equality conjecture. For any $k$,

$$c_k(\Gamma)=d_k(\Gamma).$$

The equality would give an interpretation of the higher cut numbers in terms of epimorphisms onto nilpotent quotients of free groups. The source presents this as a conjectural equality and mentions a possible approach through the Magnus expansion and the relationship between its coefficient functions and Massey products.