The 3-ball conjecture for self-affine tiles

A self-affine tile is a compact set obtained as the attractor of an expanding integral affine map with a finite digit set. Under the conditions of Theorem~, the tile is a self-affine tile whose boundary is a $2$-sphere.

3-ball conjecture. Every self-affine tile satisfying the conditions of Theorem~ is homeomorphic to a $3$-ball.

The question asks whether the spherical-boundary conclusion in Theorem~ forces the tile itself to be a $3$-ball. The authors note that this is known for individual examples via an algorithm of Conner and Thuswaldner, but they do not know how to establish it for the whole class; a self-affine Alexander horned sphere shows that a spherical boundary alone is insufficient.