Roundness of the square-lattice limit shape

Determine whether the limit shape B for the clockwise rotor walk with independent uniform initial rotors on the square lattice is a Euclidean disk.

Background

The main theorem establishes that the rescaled range converges to a deterministic compact convex set B, but does not identify the geometry of B. Numerical simulations reported in the paper suggest rotational symmetry on the square lattice and also indicate disk-shaped limits for several other lattices.

The unresolved question concerns whether the square-lattice limit shape is exactly Euclidean rather than merely convex.