  An ontological model is termed as maximally $\psi$-epistemic if the overlap
between any two quantum states is fully accounted for by the overlap of their
respective probability distributions of ontic states. However, in literature,
there exists the two different mathematical definitions (termed here as
1M$\psi$E and 2M$\psi$E) that capture the equivalent notion of maximal
$\psi$-epistemicity. In this work, we provide three theorems to critically
examine the connections between preparation non-contextuality and the
aforementioned two definitions of maximal $\psi$- epistemicity. In Theorem-1,
we provide a simple and direct argument of an existing proof to demonstrate
that the mixed state preparation non-contextuality implies the first definition
of maximal $\psi$-epistemicity. In Theorem-2, we prove that the second
definition of maximal $\psi$-epistemicity implies pure-state preparation
non-contextuality. If both the definitions capture the equivalent notion of
maximal $\psi$-epistemicity then from the aforementioned two theorems one
infers that the mixed-state preparation non-contextuality implies pure-state
preparation non-contextuality. But, in Theorem-3, we demonstrate that the
mixed-state preparation non-contextuality in an ontological model implies
pure-state contextuality and vice-versa. This leads one to conclude that
1M$\psi$E and 2M$\psi$E capture inequivalent notion of maximal
$\psi$-epistemicity. The implications of our results and their connections to
other no-go theorems are discussed.
