Existence of a continuous function nowhere differentiable along every unit speed curve

A unit speed curve is a curve parametrized by arclength, and a $C^1$ curve has a continuous first derivative. Existence conjecture. There is a continuous function

$$f\colon {\mathbb R}^d\to{\mathbb R}$$

whose restriction to any $C^1$ unit speed curve is nowhere differentiable. The main theorem establishes the analogous result when the curves also have $C^{1,\gamma}$ regularity for some $\gamma\in(0,1]$; removing this extra smoothness assumption is a natural open question.