Louder--Wilton conjecture on negative immersions and hyperbolicity

Let $G$ be a one-relator group, and let $X$ be its presentation complex. Say that $X$ has negative immersions in the sense used for presentation complexes of one-relator groups.

Louder--Wilton conjecture. Every one-relator group whose presentation complex has negative immersions is hyperbolic.

The conjecture is a geometric characterization motivated by the fact that one-relator groups with negative immersions cannot contain Baumslag--Solitar subgroups. The paper's abstract says that this conjecture is resolved by proving the assertion, so it is recorded here as solved.