Summary: In the context of maximizing cumulative dividends under barrier policies, generalized Azéma-Yor (draw-down) stopping times receive increasing attention during these past years. Based on Pontryagin’s maximality principle, we illustrate the necessity of such generalizations under the framework of spectrally negative Markov processes. Roughly speaking, starting from the explicit expression of the optimal value of discounted dividends in terms of the scale functions, we write down the optimality conditions (via Pontryagin’s principle). The use of generalized draw-downs is then quantified through a structure term (linked to the existence of non bang-bang optimal controls). We thoroughly study several classes of Lévy processes [J. Bertoin, Lévy processes. Cambridge: Cambridge Univ. Press (1996; Zbl 0861.60003); A. E. Kyprianou, Fluctuations of Lévy processes with applications. Introductory lectures. 2nd ed. Berlin: Springer (2014; Zbl 1384.60003)] constituting the usual models of insurance claims and a particular piece-wise deterministic Markov model (extending the premium rate to reserve-dependent settings). In all these models, we disprove the consistency of the aforementioned structure equation, thus denying the necessity of such generalizations. We end the paper with some heuristics on possible non-trivial cases for general Markov models.