Generic nowhere differentiability along unit speed curves

Let $U$ be a bounded open set in ${\mathbb R}^d$. Write $C(\overline U)$ for the space of continuous real-valued functions on $\overline U$, and let $G_\delta$ denote a countable intersection of open sets. Genericity conjecture. The set of functions in $C(\overline U)$ whose restriction to each $C^1$ unit speed curve is nowhere differentiable contains a dense $G_\delta$ subset. The first conjecture asks for existence on all of ${\mathbb R}^d$, whereas this conjecture asserts genericity on a bounded domain; the preceding theorem only proves the corresponding statement for curves with additional Hölder regularity of the derivative.