Which statistical hypotheses are afflicted with false confidence?

The false confidence theorem establishes that, for any data-driven, precise-probabilistic method for uncertainty quantification, there exists (both trivial and non-trivial) false hypotheses to which the method tends to assign high confidence. This raises concerns about the reliability of these widely-used methods, and shines promising light on the consonant belief function-based methods that are provably immune to false confidence. But an existence result alone leaves much to be desired. Towards an answer to the title question, I show that, roughly, complements of convex hypotheses are afflicted by false confidence.