It would take another year and a half for them to assemble a true counterexample to the periodic tiling conjecture.

D. B. A. Epstein, V. Markovic, The Logarithmic Spiral: A Counterexample to the K = 2 Conjecture, Annals of Mathematics, Second Series, Vol. 161, No. 2 (Mar., 2005 ...

The Kaplansky conjectures are three long-standing open problems on the group rings of torsion-free groups. Last week, Dr. Giles Gardam, postdoc in Mathematics Münster's topology group, announced that ...

But they’re related. Greenfeld and Tao planned to come up with a discrete counterexample to the conjecture that they could then modify to work in the continuous case as well.