Real-rootedness of arbor Ehrhart polynomials

Prove that, for every arbor t, all roots of the Ehrhart polynomial of the lattice polytope Q_t are real numbers in the interval [-1,0].

Background

The paper associates to every arbor t a lattice polytope Q_t and studies its Ehrhart polynomial E_t, which counts lattice points in integer dilates of Q_t. The authors establish recursive formulas for computing these polynomials and report that the stated root-location property has been checked computationally for all arbors of size at most 11. They leave the general real-rootedness and interval-containment assertion unresolved.

References

For any arbor $t$, all roots of the Ehrhart polynomial of $Q_t$ are real numbers in the interval $[-1,0]$.

— On posets and polytopes attached to arbors  (2503.04247 - Chapoton, 6 Mar 2025) in Section 3, subsection “Ehrhart polynomial”