One-relator hyperbolization conjecture

Let $G$ be a one-relator group whose presentation complex has negative immersions. One-relator hyperbolization conjecture. The group $G$ is hyperbolic. The conjecture would imply the conjugacy and isomorphism problems for one-relator groups with negative immersions, and the source notes that the case $\pi(w)=1$ is already known to be hyperbolic while the general conjecture remains unresolved.