On Gardner's conjecture

Gardner conjectured that if two bounded measurable sets $A,B \subset $\mathbb{R}$^n$areequidecomposablebyasetofisometries$\Gamma$ generating an amenable group then $A$ and $B$ admit a measurable equidecomposition by all isometries. Cie'sla and Sabok asked if there is a measurable equidecomposition using isometries only in the group generated by $\mathrm{\Gamma }$. We answer this question negatively.