  We find Gaussian cutoff profiles for the total variation distance to
stationarity of a random walk on a multiplex network: a finite number of
directed configuration models sharing a vertex set, each with its own bounded
degree distribution and edge probability. Further we consider the minimal total
variation distance over this space of possible doubly stochastic edge
probabilities at each point in time. Looking at all possible dynamics
simultaneously on one realisation of the random graph, we find that this
sequence of minimal distances converges in probability to the same cutoff
profile as the chain with entropy maximising transition probabilities.
