Positive expansion of products of Demazure characters into atoms

Determine whether the product of two Demazure characters expands positively into Demazure atoms.

Background

The paper studies positive combinatorial expansions involving permuted-basement Demazure atoms and Schur polynomials. Demazure characters, also called key polynomials, are a special case of the broader class of polynomials considered in the paper, and they are known to have positive expansions when multiplied by a Schur polynomial. The authors explicitly identify a more general unresolved positivity question concerning products of two characters: whether such products can be expressed as nonnegative combinations of Demazure atoms.