Conjecture on the exceptional set of nondifferentiability of the entropy function

Let $H$ be the entropy function and let ${\mathcal{E}}$ denote the exceptional set introduced in the paper. By Theorem 2, $H$ is differentiable inside each entropy plateau $(p_L,p_R)$, so the set of points where $H$ is not differentiable is contained in ${\mathcal{E}}$. Nondifferentiability conjecture. The set ${\mathcal{E}}$ is exactly the set of points where $H$ is not differentiable. This would identify the exceptional set completely; the source provides no resolution of the conjecture.