Does every torsion-free amenable group $G$ admit a system of (unitary) representations, $\mathbf R(G)$, with the following two properties:

1. $\bigcap\{\ker\pi: \pi\in\mathbf R(G)\}=1$,
2. $\pi\otimes\rho\in\mathbf R(G\times G)$, for $\pi,\rho\in\mathbf R(G)$.