Lascoux positivity of square root Demazure characters

Prove that for every square root crystal $C\subseteq RPD_{m,n}$ and every permutation $w\in S_n$, the character $ch(C_w)$ is a nonnegative integer linear combination of Lascoux polynomials.

Background

The paper constructs subsets $C_w$ of square root crystals using rectangular pipe dreams and Bruhat order, providing an analogue of Demazure crystals. It proves recursive identities for these subsets and their characters involving the $K$-theoretic divided-difference operators.

The conjectural positivity statement would extend the paper’s symmetric Grothendieck-positivity theorem to these generally nonsymmetric Demazure-type characters. The authors report computational verification for $m,n\leq 5$, except for $m=n=5$, and note that the case of the longest permutation follows from the main theorem.