The derived-series analogue for one-relator groups

Let $F$ be a free group, let $w\in F$, and let $G=F/\langle\!\langle w\rangle\!\rangle$ be the corresponding one-relator group. For a subset of $F$, write $\langle\!\langle\cdot\rangle\!\rangle$ for its normal closure, and write $G^{(\omega)}$ for the intersection of the finite derived-series terms and $G^{(\omega+1)}=[G^{(\omega)},G^{(\omega)}]$. Derived-series conjecture. There is a word $r\in F$ such that

$$w\in r[\langle\!\langle r\rangle\!\rangle,\langle\!\langle r\rangle\!\rangle]$$

and

$$G^{(\omega+1)}=G^{(\omega)}=\langle\!\langle r\rangle\!\rangle_G.$$

In particular, the maximal residually solvable quotient of $G$ is the one-relator group $F/\langle\!\langle r\rangle\!\rangle$. This conjectures that the result proved in the paper for the rational derived series extends to the usual derived series of one-relator groups; the preceding corollary establishes it when $H^2(G)=0$, while the general case remains open because local indicability of successive quotients was essential to the proofs.