Louder–Wilton conjecture on relative hyperbolicity (π(w)=2 case)

Prove that for a one‑relator group G = F/<<w>> with primitivity rank π(w) = 2, the group G is hyperbolic relative to its w‑subgroup P ≤ G.

Background

When π(w)=2, two‑generator subgroups can obstruct direct hyperbolicity. The conjecture asserts relative hyperbolicity with respect to the distinguished w‑subgroup, extending strong hyperbolic control to this borderline case.

It aligns with the hierarchical approach to one‑relator groups and would generalize many hyperbolicity results.