Continuity conjecture for the dimension of Diophantine approximation sets on smooth curves

For a smooth function $f\colon\mathbb{R}\to\mathbb{R}$, let $W_{\tau}(f)$ be the set of points defined by the simultaneous Diophantine approximation conditions used above, with infinitely many rational approximations, at exponent $\tau$. Continuity conjecture. The map

$$\tau\mapsto\dim W_{\tau}(f)$$

is continuous. This would extend the preceding result for polynomial curves, where equality with the corresponding resonant-set dimension is known at every continuity point and hence for almost every $\tau$; continuity for arbitrary smooth functions is left as a conjecture.