Real-rootedness of cyclohedron and chordal nestohedron toric g-polynomials

Prove that the g-polynomials of the n-dimensional cyclohedron and chordal nestohedra are real-rooted.

Background

The paper proves that each toric g-contribution polynomial g_{n,j}(x) is real-rooted and expresses the toric g-polynomial of an n-dimensional simple polytope as a nonnegative gamma-vector-weighted linear combination of these contribution polynomials. The cited conjecture concerns whether this additional real-rootedness property holds for the toric g-polynomials associated with n-dimensional cyclohedra and chordal nestohedra.

The gamma-vectors of both families are nonnegative, so the conjecture would follow from a sufficiently strong interlacing property among the contribution polynomials. The paper does not establish this conjecture; instead, it later formulates a broader interlacing conjecture whose validity would imply it.