  We study quantization ambiguities in loop quantum cosmology that arise for
space-times with non-zero spatial curvature and anisotropies. Motivated by
lessons from different possible loop quantizations of the closed
Friedmann-Lemaitre-Robertson-Walker cosmology, we find that using open
holonomies of the extrinsic curvature, which due to gauge-fixing can be treated
as a connection, leads to the same quantum geometry effects that are found in
spatially flat cosmologies. More specifically, in contrast to the quantization
based on open holonomies of the Ashtekar-Barbero connection, the expansion and
shear scalars in the effective theories of the Bianchi type II and Bianchi type
IX models have upper bounds, and these are in exact agreement with the bounds
found in the effective theories of the Friedmann-Lemaitre-Robertson-Walker and
Bianchi type I models in loop quantum cosmology. We also comment on some
ambiguities present in the definition of inverse triad operators and their
role.
