Interlacing of the full toric g-contribution polynomial sequence

Establish that the polynomial sequence (g_{n,0}(x), g_{n,1}(x), ..., g_{n,\lfloor n/2\rfloor}(x)) is interlacing for every integer n greater than or equal to zero.

Background

The main theorem proves real-rootedness of every toric g-contribution polynomial g_{n,j}(x) and establishes interlacing in the n-index: g_{n,j}(x) interlaces g_{n+1,j}(x), as well as g_{n+1,j+1}(x). The unresolved question asks for the stronger assertion that, for each fixed n, the entire sequence indexed by j is interlacing.

The author reports a Mathematica verification of this property for all n less than or equal to 120, but explicitly leaves the general statement as a conjecture. If true, the result would imply real-rootedness of toric g-polynomials for every n-dimensional simple polytope with a nonnegative gamma-vector, including the cyclohedron and chordal nestohedron conjecture stated immediately beforehand.

References

Although this verification is finite, the data suggests that the same property may persist for all $n$. Athanasiadis further suggested that we formulate the following conjecture. The polynomial sequence $(g_{n,0}(x),g_{n,1}(x),\ldots,g_{n,{n/2}(x))$ is interlacing for every $n\ge 0$.

— The real-rootedness of the toric $g$-contribution polynomials  (2609.01086 - Xiao, 1 Sep 2026) in Section 4, Conclusion; Conjecture labeled Conjg