Minimal entropy production due to constraints on rate matrix dependencies in multipartite processes

I consider multipartite processes in which there are constraints on each subsystem's rate matrix, restricting which other subsystems can directly affect its dynamics. I derive a strictly nonzero lower bound on the minimal achievable entropy production rate of the process in terms of these constraints on the rate matrices of its subsystems. The bound is based on constructing counterfactual rate matrices, in which some subsystems are held fixed while the others are allowed to evolve. This bound is related to the "learning rate" of stationary bipartite systems, and more generally to the "information flow" in bipartite systems.