Characterization of the FLTZ mirror of the structure sheaf for arbitrary (non-proper) toric varieties

Develop an explicit characterization of the constructible sheaf corresponding, under the FLTZ mirror functor, to the structure sheaf \mathcal{O} for arbitrary (not proper) toric varieties; specifically, identify \Phi_{FLTZ}(\mathcal{O}) beyond the proper case where the mirror is the skyscraper sheaf at the identity.

Background

Lemma O shows that for DM toric stacks with proper coarse moduli space, any monoidal functor from coherent sheaves to constructible sheaves sends the structure sheaf \mathcal{O} to the skyscraper sheaf at 0, Z_0.

The author remarks that for arbitrary (non-proper) toric varieties, the explicit description of the mirror constructible sheaf corresponding to \mathcal{O} under the FLTZ functor is unknown, highlighting a gap in the monoidal understanding of toric mirror symmetry outside the proper setting.