Freeness of Hom-spaces in the categorification of extremal weight modules

Determine whether the Hom-spaces of the categories \(\dot V^\lambda_\mu\) are free in general over the relevant Henselian local rings.

Background

The paper constructs a symmetric bilinear form on the extremal weight module and discusses its dependence on the choice of base field and quiver Hecke datum. Independence follows under a freeness hypothesis for the Hom-spaces of the categorifying categories $\dot V^\lambda_\mu$. The paper explicitly leaves unresolved whether this freeness property holds in general.