Shifted positivity for queer square root crystals

Prove the shifted analogue of the character theorem for normal square root crystals associated with the queer Lie superalgebra $q_n$, namely that their characters are positive in the relevant $K$-theoretic Schur $P$- or shifted Grothendieck basis.

Background

The paper notes that a theory of normal square root crystals associated with the queer Lie superalgebra $q_n$ already exists. These objects are related to $q_n$-crystals in a manner analogous to the relationship between the paper’s square root crystals and $gl_n$-crystals.

The paper’s main theorem establishes Grothendieck positivity for characters of finite normal square root crystals of type $gl_n$. The unresolved conjectural extension replaces Grothendieck positivity by positivity in a shifted, $K$-theoretic Schur $P$-function basis for the queer setting.