The adjacency-preservation conjecture for substitution systems

A substitution system is a tiling system equipped with tile substitutions; it preserves adjacency when, for any two tiles $t,t'$ belonging to distinct supertiles $s,s'$, the tiles share an edge if and only if the supertiles do, and the same holds for $n$th-order supertiles for every $n$. Adjacency-preservation conjecture. For any substitution system, there exists $n\geqslant 0$ such that deflating the tiles' edges $n$ times makes it adjacency-preserving. This would extend the finite-state transducer construction beyond spurless systems to arbitrary substitution systems; it is verified for the systems discussed in the paper, but no general proof is given.