Cyclic orders and actions of Leary--Minasyan groups on coarse $\mathrm{P}\mathrm{D}(n)$ spaces

We show that for $n\ge 3$, there are torsion-free CAT(0) groups with visual boundaries that embed into $S^n$ but which are not virtually the fundamental group of a compact aspherical $n+1$-manifold. The groups are the CAT(0) and not bi-automatic groups constructed previously by Leary and Minasyan. The obstruction comes from analyzing certain cyclic orders on the boundary of the Bass-Serre tree, in the same manner as Kapovich-Kleiner ruled out actions of Baumslag-Solitar groups on coarse $\mathrm{P}\mathrm{D}(3)$ spaces.