The studies of multiplex networks are increasingly popular in recent years. Modeling multiple complex systems as a multiplex network has refreshed our understanding about the structure and dynamics of various real-world systems. As an important variant of the voter models, belief formation dynamics such as the asynchronous belief percolation (ABP) model has attracted much attention from statistical physics and network science communities. Existing studies of the ABP model mainly focus on the applications to single networks, whereas how the structure of multiplex networks affects its dynamical behavior is still not well understood. To close this gap, we propose a multi-scale ABP model that takes into account the differential velocities of belief propagation at different subnetworks within the multiplex network. Using extensive computer simulations, we find that (i) increasing the degree correlation between subnetworks can promote nodes with minority belief to form stable clusters and (ii) minority nodes require less initial supports to survive in multiplex networks with respect to single networks. Our conclusion is robust against the detailed topology of the subnetworks that constitute the multiplex network.