Nonnegative SEM expansions

Prove or refute the conjecture that the SEM expansion of a Schubert polynomial has all nonnegative coefficients only when the Schubert polynomial is a single standard elementary monomial.

Background

The paper characterizes permutations whose Schubert polynomials are single SEMs as precisely those avoiding the patterns 312 and 1432. It then proposes a stronger coefficient-positivity conjecture: among all Schubert polynomials, coefficientwise nonnegative SEM expansions should occur only in the single-SEM cases. The authors note that 312-avoidance follows from an earlier argument, leaving 1432-avoidance as the remaining issue.

References

We guess that a stronger fact is true:

The SEM expansion of $\mathfrak{S}_w$ has all nonnegative coefficients only when $\mathfrak{S}_w$ is a single SEM.

— Single-SEM Schubert Polynomials  (2503.03903 - Woodruff, 5 Mar 2025) in Section 3, immediately following the proof of Theorem 1.1