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

 1. $\cap\{\ker\pi: \pi\in\mathbf R(G)\}=0$, 
 2. $\pi\otimes\rho\in\mathbf R(G\times G)$, for $\pi,\rho\in\mathbf R(G)$.