On the cancellation problem for algebraic tori

We address a variant of Zariski Cancellation Problem, asking whether two varieties which become isomorphic after taking their product with an algebraic torus are isomorphic themselves. Such cancellation property is easily checked for curves, is known to hold for smooth varieties of log-general type by virtue of a result of Iitaka-Fujita, and more generally for varieties which are not dominantly covered by images of the punctured affine line. We show in contrast that cancellation fails in general in every dimension bigger or equal to two.