Authors’ conjecture: existence of a non–virtually free group with decidable domino problem

Construct a finitely generated group that is not virtually free and has a decidable domino problem, thereby refuting the Ballier–Stein conjecture. Specifically, improve the tower-of-HNN-extension construction in the paper to produce such a group.

Background

The authors present a non–virtually free group with decidable snake tiling. This forces either the Ballier–Stein conjecture to be false or a separation in difficulty between the domino and snake problems. They explicitly conjecture that the former holds and that their construction can be strengthened to yield a non–virtually free group with decidable domino problem.