Does κ>1 require symmetry of the Markov chain?

Determine whether the occurrence of a random attractor with cardinality κ greater than 1 for some random dynamical system representation of an irreducible and positive recurrent Markov chain necessarily requires that the Markov chain admits a symmetry.

Background

Empirically, all presented examples yield either κ=1 or κ>1 in the presence of apparent symmetries (e.g., symmetric partitions with equal stationary mass). Proposition 4.6 constrains κ by requiring a partition of X into κ sets of equal π-mass.

Despite these hints, the authors note that it remains unclear whether κ>1 can only arise when the chain possesses some form of symmetry, leaving open a structural characterization of when larger attractors are possible.