  We interpret the Hilbert entropy of a convex projective structure on a closed
higher-genus surface as the Hausdorff dimension of the non-differentiability
points of the limit set in the full flag space $\mathcal F(\mathbb R^3)$.
Generalizations for regularity properties of boundary maps between locally
conformal representations are also discussed. An ingredient for the proofs is
the concept of hyperplane conicality that we introduce for a $\theta$-Anosov
representation into a reductive real-algebraic Lie group $G$. In contrast with
directional conicality, hyperplane-conical points always have full mass for the
corresponding Patterson-Sullivan measure.
