Can a dialetheist stay regular?

For a dialetheist, it is rational to believe that true contradictions exist. However, we argue that a dialetheist faces a complex dilemma given some bridge principles for rational beliefs that connect possibility and probability, including the so-called ‘Regularity Principle’. Either her belief is not doxastically possible even for her, or she must assign positive credence to the proposition that dialetheia exists. The former makes her belief prima facie self-defeating. The latter seems to compel her to choose between several more fine-grained options, each of which seems undesirable for a regular and rational probability theory. It is unclear how a dialetheist would stay rationally regular, so we pose this challenge, seeking a response from a dialetheist.