The Kaplansky conjecture and Kaplanski-Kadison conjecture are classical conjectures about non existence of non-trivial idempotents in group algebra $\mathbb{C}\Gamma $ or the reduced group algebra $C_{\mathrm{red}}^*\Gamma$ where $\Gamma$ is a torsion free group.

On the other hand, the value of trace of an idempotent plays a crucial role in investigation of idempotent problem in certain $C^*$-algebra or group algebra. The rationality or integrality of $\tau(e)$ or the range of change of $\tau(e)$... etc. See Idempotents in complex group rings: theorems of Zalesskii and Bass revisited

Now we notice that trace is a cocycle of the Hochschild complex.

So a natural question is that what type of other higher dimensional cocycles have been examined in the context of idempotent problems for group algebra or reduce group algebras.

As a related point we know from page 21 of Non-Commutative Geometry that $\tau(e,e,e)$ is constant on every curve of idempotents where $\tau$ is a cyclic 2-cocycle.

But is there any precise case of torsion-free group $\Gamma$ for which the standard trace does not work but a higher order cocycle works to prove that the corresponding algebra does not have any non trivial idempotent?