Nonnegative SEM expansion conjecture for Schubert polynomials

Let $w$ be a permutation, let $\mathfrak{S}_w$ denote its Schubert polynomial, and consider its expansion in the SEM basis. A polynomial is a single SEM if it is one basis element of this expansion. Nonnegative SEM expansion conjecture. The SEM expansion of $\mathfrak{S}_w$ has all nonnegative coefficients only when $\mathfrak{S}_w$ is a single SEM. The preceding argument proves that violating the stated Lehmer-code rules prevents a Schubert polynomial from being a single SEM; the stronger nonnegativity assertion is presented as a conjectural extension and is not resolved here.