  Von Neumann stability analysis of quantum difference equations in loop
quantized spacetimes has often proved useful to understand viability of
quantizations and whether general relativistic description is recovered at
small spacetime curvatures. We use this technique to analyze the infra-red
behavior of quantum Hamiltonian constraint in recently explored modifications
of loop quantum cosmology: mLQC-I and mLQC-II, for the spatially flat FLRW
model. We investigate the behavior for $\mu_o$ scheme, where minimum area of
loops in quantization procedure does not take physical metric in to account,
and the $\bar \mu$ scheme where quantization procedure uses physical metric.
The fate of stability of quantum difference equations is tested for massless
scalar field as well as with inclusion of a positive cosmological constant. We
show that for mLQC-I and mLQC-II, difference equation fails to be von Neumann
stable for the $\mu_o$ scheme if cosmological constant is included signaling
problematic behavior at large volumes. Both of the modified loop quantum
cosmologies are von Neumann stable for $\bar \mu$ scheme. In contrast to
standard loop quantum cosmology, properties of roots turn out to be richer and
intricate. Our results demonstrate the robustness of $\bar \mu$ scheme (or
`improved dynamics') in loop quantization of cosmological spacetimes even when
non-trivial quantization ambiguities of Hamiltonian are considered, and show
that $\mu_o$ scheme is non-viable in this setting.
