Modelling higher-order vagueness: columns, borderlines and boundaries

According to columnar higher-order vagueness (CHOV), all orders of vagueness coincide: any borderline case is a borderline borderline case, and a third-order borderline case, etc. Bobzien (2015) has worked out many details of such a theory and models it with a modal logic closely related to S4. I take up a range of questions about the framework and argue that it is not suitable for modelling the structure of vagueness and higher-order vagueness.