Louder–Wilton conjecture on relative hyperbolicity at primitivity rank 2

Prove that for any one-relator group $G = F/{w}$ with primitivity rank $\pi(w)=2$, the group $G$ is hyperbolic relative to its unique $w$-subgroup $P \leqslant G$.

Background

At primitivity rank two, one-relator groups have a distinguished two-generator $w$-subgroup capturing non-free rank-two substructures. Relative hyperbolicity to this subgroup would extend hyperbolicity results beyond the 2-free cases.