I think the property you formulate always holds and is implied by Bourbaki's discussion of "saturated" sets of weights, as applied here to minuscule or quasi-minuscule weights (references given in recent related questions on MO).   The main point is that the Weyl group orbit of the highest weight exhausts all weights (except possibly 0).  So the root strings involved in the weight diagram all have length 0 or 1, from the saturation property.   This has to be adapted to the quasi-minuscule case, but the idea should be the same as for minuscule weights.