Cubulating a free-product-by-cyclic group

Let $G=H_1\ast \mathrm{.}\mathrm{.}\mathrm{.}\ast H_k\ast F_r$ be a torsion-free group and $\phi$ an automorphism of $G$ that preserves this free factor system. We show that when $\phi$ is fully irreducible and atoroidal relative to this free factor system, the mapping torus $\mathrm{\Gamma }=G{⋊}_{\phi }\mathbb{Z}$ acts relatively geometrically on a hyperbolic CAT(0) cube complex. This is a generalisation of a result of Hagen and Wise for hyperbolic free-by-cyclic groups.