Are hyperbolic one-relator groups virtually compact special?

Theorem 1.1: every word-hyperbolic one-relator group is virtually compact special. Theorem 1.2: every hyperbolic ascending mapping torus of an injective endomorphism of a finitely generated free group is virtually compact special. With Kielak-Linton, every hyperbolic one-relator group is virtually free-by-cyclic (kernel possibly of infinite rank in the subgroup-closed sense, with a finite-index embedding in a free-by-$\mathbb Z$ group with finitely generated kernel), which is Wise's Conjecture 17.8 for this class, and every hyperbolic one-relator group is residually finite and linear over $\mathbb Z$. It says nothing about non-hyperbolic one-relator groups. Consistent with the release's non-residually-finite hyperbolic group (catalog entry on residual finiteness), which is not one-relator.

Theorem 1.1: every word-hyperbolic one-relator group is virtually compact special. Theorem 1.2: every hyperbolic ascending mapping torus of an injective endomorphism of a finitely generated free group is virtually compact special. With Kielak-Linton, every hyperbolic one-relator group is virtually free-by-cyclic (kernel possibly of infinite rank in the subgroup-closed sense, with a finite-index embedding in a free-by-$\mathbb Z$ group with finitely generated kernel), which is Wise's Conjecture 17.8 for this class, and every hyperbolic one-relator group is residually finite and linear over $\mathbb Z$. It says nothing about non-hyperbolic one-relator groups. Consistent with the release's non-residually-finite hyperbolic group (catalog entry on residual finiteness), which is not one-relator.