Necessity of negative returns for steady-state multiplicity

Prove that a negative return on saving, equivalently a rental rate satisfying r<δ in the canonical Aiyagari model, is necessary for the existence of multiple steady states.

Background

The paper constructs an analytic example of the canonical Aiyagari model with at least three steady states. The construction relies on bounds for stationary capital supply when the gross return on saving is below one, which is equivalent to a rental rate below the depreciation rate (r<δ).

The authors explicitly conjecture that this negative-return region is not merely useful for their construction but is necessary for multiplicity in the model. They do not establish this necessity result, leaving it as an unresolved theoretical question.