  It has been well established that quantum mechanics (QM) violates Bell
inequalities (BI), which are consequences of local realism (LR). Remarkably QM
also violates Leggett inequalities (LI), which are consequences of a class of
nonlocal realism called crypto-nonlocal realism (CNR). Both LR and CNR assume
that measurement outcomes are determined by preexisting objective properties,
as well as hidden variables (HV) not considered in QM. We extend CNR and LI to
include the case that the measurement settings are not externally fixed, but
determined by hidden variables (HV). We derive a new version of LI, which is
then shown to be violated by entangled $B_d$ mesons, if
charge-conjugation-parity (CP) symmetry is indirectly violated, as indeed
established. The experimental result is quantitatively estimated by using the
indirect CP violation parameter, and the maximum of a suitably defined relative
violation is about $2.7\%$. Our work implies that standard model (SM) of
particle physics violates CNR. Our LI can also be tested in other systems such
as photon polarizations.
