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EDIT: Reading the question more carefully, I think the difference between the highest weight and an arbitrary Weyl group conjugate will almost never be a single root. (What's true is that the difference between "adjacent" weights in that orbit across a single reflecting wall will be 0 or else a root. The adjoint representation in type $E_8$ illustrates this behavior. The saturation property in Bourbaki implies here that weight strings between such adjacent weights are of length 0 or 1.)