  Leggett Garg inequalities (LGIs) test the incompatibility between the notion
of macrorealism and quantum mechanics. For unitary dynamics, the optimal
quantum violation of an LGI is constrained by the L\"uders bound. However, the
LGIs does not provide the necessary and sufficient for macrorealism. A suitably
formulated set of no-signaling in time (NSIT) conditions along with the
arrow-of-time (AOT) condition provides the same. In this paper, we study two
formulations in the three-time LG scenario, viz., the standard LGIs and the
recently formulated variant of LGIs when the system evolves under
$\mathcal{PT}$-symmetric Hamiltonian. We first demonstrate that the quantum
violations of both forms of LGIs exceed their respective L\"uders bounds and
can even reach their algebraic maximum. We further show that for the case of
standard LGI, the violation of L\"uders bound can be obtained when both NSIT
and AOT conditions are violated. Interestingly, for the case of a variant of
LGI, for suitable choices of relevant parameters, the quantum violation can
even be obtained when only the AOT is violated but all NSIT conditions are
satisfied. This feature has not hitherto been demonstrated. We discuss the
further implication of our study.
