Cantor attractor conjecture for random piecewise isometries

Let $S^1$ be the circle, let $A$ be an interval, and let $X(w)$ denote the topological attractor

$$X(w)=\bigcap_{n\in\mathbb{N}} \overline{F_{w,\mathrm{rev}}^n(S^1)},$$

where $F_{w,\mathrm{rev}}^n=f_{w_1}\circ\dots\circ f_{w_n}$ is the reversed-order composition associated with the random sequence $w$.

Cantor attractor conjecture. If $A$ is an interval, the topological attractor $X(w)$ is almost surely a Cantor set.

This predicts a discrepancy between the topological attractor and the measurable limit of the pushed-forward Lebesgue measures: although the latter converges almost surely to a point mass, the former remains a totally disconnected perfect set. The source presents this as a prediction based on numerical simulations and rough heuristic arguments; no resolution is supplied.