  In the context of quantum objectivity, a standard way to quantify the
classicality of a state is via the mutual information between a system and
different fractions of its environment. Many of the tools developed in the
relevant literature to quantify quantum objectivity via quantum mutual
information rely on the assumption that information about the system leaks
symmetrically into its environment. In this work, we highlight the importance
of taking this assumption into account, and in particular, analyse how taking
non-averaged quantum mutual information as a quantifier of quantum objectivity
can be severely misleading whenever information about the system is encoded
into the environment in a non-homogeneous way. On the other hand, the averaged
mutual information always provides results with a clear operative
interpretation.
