Canonical-transversal preservation conjecture for homeomorphisms of tiling spaces

Let $\Omega$ and $\Omega'$ be aperiodic, repetitive tiling spaces with finite local complexity, and let $h:\Omega\rightarrow\Omega'$ be a homeomorphism. A canonical transversal $\Xi$ is a transversal obtained by requiring a tile (or marked cell) of each tiling to lie at the origin.

Canonical-transversal preservation conjecture. There is a homeomorphism $h'$ isotopic to $h$ such that, for any canonical transversal $\Xi$ in $\Omega$, $h'(\Xi)$ is a canonical transversal in $\Omega'$.

The question concerns whether arbitrary homeomorphisms between aperiodic repetitive FLC tiling spaces can be adjusted within their isotopy class to preserve all canonical transversals. The supplied context presents this as an unresolved question rather than reporting a proof or disproof.