Determine whether an ASPmr tile set exists

Determine whether there exists an aperiodic tile set consisting of two non-congruent tile types that generates non-periodic tilings in which the densities of the two tile types are mutually reversed, namely an \(ASPmr\{\text{A-tile},\text{B-tile}\}\).

Background

The paper defines $ASPmr\{\text{A-tile},\text{B-tile}\}$ as an aperiodic two-tile set, without matching rules, whose generated non-periodic tilings exhibit mutually reversed densities of the two tile types. The hat–turtle pair has the density-reversal property but also admits periodic tilings, so it is not an example. The four pairs involving AH-, BH-, AT-, and BT-tiles are presented as possible candidates, but their aperiodicity is not established.