Relative hyperbolization conjecture for primitivity rank two one-relator groups

Let $F$ be a free group, let $w\in F\setminus\{1\}$ have primitivity rank $\pi(w)=2$, and let $G=F/\langle\!\langle w\rangle\!\rangle$. Let $P<G$ be the peripheral subgroup supplied by the preceding result, namely a two-generator one-relator subgroup such that every two-generator subgroup of $G$ is either free or conjugate into $P$. Relative hyperbolization conjecture. The group $G$ is hyperbolic relative to $P$. This is proposed as a counterpart to the one-relator hyperbolization conjecture; the source gives no resolution and identifies the primitivity-rank-two case as the remaining case of interest after the torsion case.