Uniform conjugacy invariance of shifts of finite type

A shift space is a closed, shift-invariant subset of a full shift, and an SFT (shift of finite type) is a shift space defined by finitely many forbidden finite patterns. Two shift spaces are uniformly conjugate when they are related by a uniformly continuous conjugacy.

Uniform conjugacy invariance conjecture. Any shift which is uniformly conjugate to an SFT is itself an SFT.

This holds for shift spaces over the monoids $\mathbb{N}$ and $\mathbb{Z}$, but the proof in those cases does not extend to the general context. A positive answer to the paper's shadowing-property problem would also imply this conjecture.