Generating-set independence of snake tiling problems

Determine whether the decidability and definitions of snake tiling problems (weak/strong, directed/undirected, infinite/ouroboros) for a given finitely generated group are independent of the choice of finite generating set.

Background

The authors discuss multiple variants of snake tiling problems and note that, unlike the domino problem, it is not known whether snake problems are invariant under changing the generating set. They reiterate this uncertainty in the appendix while introducing K-snakes as a small step toward geometric invariance.