Nonnegativity conjecture for Schubert expansions of transparent diagrams

Let $D$ be a diagram, and write its flagged Schur character in the Schubert-polynomial basis as

$$\mathfrak{S}_D=\sum_w c^w_D\,\mathfrak{S}_w,$$

where the sum is over all permutations and $c^w_D$ are integers. A diagram is called transparent when it has the transparency property defined in the paper. Nonnegativity conjecture. When $D$ is transparent, the coefficients $c^w_D$ are nonnegative:

$$c^w_D\geq 0$$

for every permutation $w$. This conjecture predicts Schubert positivity for the characters of flagged Schur modules associated with transparent diagrams; the source motivates it through computer experimentation, while noting that related classes such as translucent and percentage-avoiding diagrams can have negative coefficients.