gauge theory and its zero-dimensional reduction, which is known as [*Eguchi-Kawai equivalence*]{} [@Eguchi:1982nm]. Here, we consider the $SU(N)$ gauge theory (YM) on $D$-dimensional periodic lattice with the Wilson’s plaquette action $$S_W=-\beta N\sum_x\sum_{\mu\neq\nu}{\rm Tr}
~U_{\mu}(x)U_{\nu}(x+\hat{\mu})U^\dagger_{\mu}(x+\hat{\nu})U^\dagger_{\nu}(x),$$ where $U_{\mu}(x)\ (\mu=1,..., D)\in SU(N)$ are link variables and $\beta$ is the inverse of the bare ’t Hooft coupling. In the large-$N$ limit the space-time degrees of freedom can be neglected, and then this theory can be equivalent to a model defined on a single hyper-cube, $$S_{EK}=-\beta N\sum_{\mu\neq\nu}{\rm Tr}
~U_{\mu}U_{\nu}U^\dagger_{\mu}U^\dagger_{\nu},$$ which is called the Eguchi-Kawai model (EK model). The equality was shown by observing that the Schwinger-Dyson equations for Wilson loops (loop equations) in both theories are the same. In the EK model the loop equations can naively have open Wilson line terms, which do not exist in the original gauge theory side due to the gauge invariance. Therefore we need to assume that the global $\mathbb{Z}_N^D$ symmetry $$U_\mu\to e^{i\theta_\mu}U_\mu,$$ which eliminates the non-zero expectation value of the open Wilson lines, is not broken spontaneously. However, soon after the discovery of the equivalence, it was found that the $\mathbb{Z}_N^D$ symmetry is actually broken for $D>2$ in the weak coupling region [@Bhanot:1982sh]. Although the naive EK equivalence 