Ouroboros snake problems on non–virtually free groups

Ascertain whether there exists a finitely generated group that is not virtually free for which at least one, or all, ouroboros (closed-loop) snake tiling problems are decidable.

Background

The paper proves decidability for several variants of snake tiling on a specific non–virtually free group but cannot settle the ouroboros variants (closed snakes). The authors pose an explicit open question about the existence of non–virtually free groups with decidable ouroboros problems.