Finiteness or infinitude of nonanalytic points

Determine whether the intermediate dimension curve d_\theta of the delayed-contraction Cantor set E_{\beta,M} has infinitely many nonanalytic points in the parameter interval (0,1).

Background

The paper proves that, on every interval [\varepsilon,1] with \varepsilon>0, the intermediate dimension curve d_\theta is real analytic on the open intervals of a finite partition and has at least one nonanalytic transition. It also establishes local finiteness of possible nonanalytic points away from \theta=0.

The global question left unresolved is whether the curve nevertheless has infinitely many nonanalytic points as \theta ranges over (0,1), potentially accumulating only at the endpoint \theta=0.