No-slip rolling realization of the equidistant separating curve

Determine whether, for every connected bounded subset A of the Euclidean plane R^2 and every ε > 0, there exists a motion of a circle (wheel) of radius ε lying in the plane such that the circle continuously touches A without intersecting A, the center of the circle traces a simple closed curve that bounds the region containing A, and the motion occurs without slipping along the boundary of A throughout the trajectory.

Background

The paper proves that for a connected bounded set A ⊂ R^2 and any ε > 0, the ε-boundary S_ε(A) contains a simple closed curve that bounds the region containing the open ε-neighborhood O_ε(A). This admits a mechanical interpretation: one can place a wheel of radius ε in the plane and move it so that it always touches A, does not intersect A, and the center traces a simple closed curve enclosing A.

While the existence of such a curve and a touching motion is established, the authors explicitly state that it is unknown whether this motion can always be realized as pure rolling without slipping. This raises a concrete kinematic realization question tied to the geometric existence result for equidistant level sets.