A torsion-free algebraically C*-unique group

Let $p$ and $q$ be multiplicatively independent integers. We show that the complex group ring of $\mathbb{Z}[\frac{1}{pq}]⋊{\mathbb{Z}}^2$ admits a unique ${\mathrm{C}}^{\ast }$-norm. The proof uses a characterization, due to Furstenberg, of closed $\times p-$ and $\times q-$invariant subsets of $\mathbb{T}$.