Gersten's distortion–divergence conjecture for normal free subgroups

Let $F$ be a finitely generated normal free subgroup of a group $G$, and suppose that the quotient $G/F$ is an infinite cyclic group.

Gersten's conjecture. The distortion of $F$ in $G$ is equivalent to the divergence of $G$.

This conjecture concerns the relationship between subgroup distortion and ambient-group divergence, particularly for normal free subgroups of groups with cyclic quotient. The source presents it as a conjecture raised by Gersten; its resolution is not established in the supplied text.