  In a recent paper, Engle, Hanusch and Thiemann showed that there is a unique
state on the reduced holonomy-flux $\ast$-algebra of homogeneous isotropic loop
quantum cosmology, that is invariant under residual diffeomorphims. This result
has been claimed to be true both for the Ashtekar-Bojowald-Lewandowski
framework and for that introduced by the present author. Unfortunately, the
uniqueness proof relies on an incorrect argument which spoils the second case.
In our short note, we are going to patch this issue, this way keeping the nice
uniqueness result in both cases. Moreover, we will even extend the underlying
operator algebraic statements as this might help later for studying
higher-dimensional models.
