Finite polynomial convexity of smooth simple Jordan arcs

Determine whether every smooth simple Jordan arc γ ⊂ C is finitely polynomially convex; that is, ascertain whether there exists a finite integer d such that the classical polynomial hull P(γ) equals the d-polynomial hull P_d(γ).

Background

The paper introduces the notion of a set being finitely polynomially convex if its classical polynomial hull coincides with a d-polynomial hull for some finite degree d. While various examples and partial results are presented, a general characterization remains elusive.

Example 19 shows that certain rectifiable (non-smooth) arcs are not finitely polynomially convex, leaving open the case of smooth simple Jordan arcs, which is posed explicitly as a question.