Cubulating mapping tori of some polynomial growth free group automorphisms

Let $F$ be a finite-rank free group and let $\Phi\in\mathrm{Out}(F)$ have polynomial growth. Let $G=F\rtimes_\Phi\mathbb{Z}$. We give sufficient conditions on $\Phi$ that ensure $G$ acts freely on a CAT(0) cube complex. For $d=1$, the class of $G$ that we cubulate strictly contains tubular free-by-cyclic groups, which were cubulated by Button. For $d>1$, we cubulate $G$ provided, for instance, the linear-growth mapping tori contained in $G$ are tubular and $G$ satisfies a condition on intersections of certain centralisers. These conditions are satisfied when the growth rate of $\Phi$ is as large as possible for $F$. Using this, we show that for any fixed $F$, a random unipotent polynomially growing automorphism $\Phi$ has cubulated mapping torus. We do not work directly with relative train tracks, but rely on them via the cyclic hierarchy from work of Macura in the superlinear case and the splitting over $\mathbb{Z}^2$ subgroups from from work of Andrew-Martino and Dahmani-Touikan in the linear case. Our proof relies on cubical small-cancellation theory to obtain free actions on CAT(0) cube complexes for groups admitting suitable acylindrical cyclic hierarchies whose bottom-level vertex groups are cubulated; this technical result is of independent interest.