Ballier and Stein's conjecture on the domino problem

Let $G$ be a finitely generated group. Recall that $G$ is virtually free if it has a finite-index free subgroup. Ballier and Stein's conjecture. The group $G$ has decidable domino problem if and only if it is virtually free.

This conjecture proposes that virtual freeness exactly characterizes finitely generated groups with decidable domino problem. The forward implication remains open as of 2023, despite substantial progress toward its resolution, including the geometric nature of the domino problem among finitely presented groups.