Violation of local detailed balance upon lumping despite a clear timescale separation

Lumping microstates is a coarse-graining paradigm in stochastic thermodynamics that supposedly gives rise to local detailed balance in the presence of a timescale separation. It is explained why lumped dynamics may generally violate local detailed balance even in the presence of a clear timescale separation. Conversely, detailed balance can be restored, even in the absence of a timescale separation, if the coarse graining is carried out as milestoning.