Uniqueness of the IFS from the neighbor automaton

Let a self-similar crystallographic or self-affine tile be given, together with its iterated function system and neighbor automaton. The combinatorial structure of the neighbor automaton records the transitions between neighboring pieces.

Uniqueness conjecture. For any self-similar crystallographic or self-affine tile, the combinatorial structure of the neighbor automaton uniquely determines the iterated function system.

The claim proposes that the combinatorial data of the neighbor automaton retain enough information to reconstruct the underlying geometric system, extending the uniqueness observed in the preceding examples. The source provides no resolution or further qualification of this general assertion.