<p>While the invariant energy quadratization (IEQ) approach has become a popular framework for constructing energy-stable schemes for gradient flow models, it remains theoretically unclear whether such schemes can rigorously preserve the discrete maximum bound principle (MBP) when applied to the Allen-Cahn equation. Although numerical evidence suggests that standard IEQ schemes often appear to satisfy the MBP in practice, no rigorous proof has been established to date. In this paper, we address this open problem by proposing and analyzing the first fully discrete, second-order accurate IEQ-based scheme that provably satisfies the discrete MBP. The key innovation lies in a new treatment of the nonlinear coefficient associated with the auxiliary IEQ variable. Instead of relying on conventional second-order extrapolation, we introduce a time-averaging strategy that blends a first-order IEQ scheme with the previous time-level solution. In addition, a carefully designed linear stabilizing term plays a pivotal role in enabling the theoretical analysis. These two components are essential in establishing both the discrete MBP and energy stability—two properties that, to our knowledge, had not been simultaneously achieved within the original IEQ framework. The temporal discretization combines the Crank-Nicolson and Euler methods, while the spatial discretization employs a standard finite difference (FD) method. We further establish optimal-order <i>L</i>²-error estimates and present a set of numerical examples that confirm the theoretical results and demonstrate the accuracy and robustness of the proposed scheme.</p>