Conjugacy invariance of finitely defined shifts

A shift space is a closed, shift-invariant subset of a full shift, and an FDS (finitely defined shift) is a shift space defined by finitely many forbidden patterns in the sense developed in the paper. Two shift spaces are conjugated when there is a shift-commuting homeomorphism between them.

Conjugacy invariance conjecture for FDSs. A shift space which is conjugated to an FDS is itself an FDS.

This conjecture would give a negative answer to the paper's problem asking whether a sofic shift that is not an FDS can be conjugate to an FDS. The source provides no resolution in the general setting.