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I don't see how your restriction captures the requirement you stated. In any case, it does not preserve integrality. Consider a case where customers A and B (each with 1 unit demand) can both be served by depots 1 and 2, but you require the flow from 1 to A be the maximum of the flows out of 1. There are two extreme points of the set of feasible solutions, of which one is non-integral (each depot sends $1/2$ unit to each customer), and this may well be the optimal solution.