Local axioms characterizing normal square root crystals

Determine local axioms in the style of Stembridge’s axioms that characterize exactly which square root crystals are normal.

Background

Normal square root crystals are combinatorial structures modeled on tensor powers of the standard square root crystal. The paper proves that finite normal square root crystals have characters that expand positively in symmetric Grothendieck polynomials, but it does not provide an intrinsic local characterization of normality analogous to the Stembridge axioms for normal $gl_n$-crystals.

The unresolved problem is to identify local conditions on the crystal graph, edge labels, weights, and operators that are necessary and sufficient for a square root crystal to be normal.