Ballier–Stein conjecture on the domino problem

Let $G$ be a finitely generated group. The domino problem asks whether a given finite set of Wang tiles admits a tiling of $G$. Ballier–Stein conjecture. $G$ has undecidable domino problem if and only if $G$ is not virtually free. The conjecture is stated as known, roughly, for the same classes of groups covered by the preceding periodicity conjectures, but the general claim remains open in the supplied text.