Convergence of spectral truncations for compact metric groups

We consider Gromov-Hausdorff convergence of state spaces for spectral truncations of a compact metric group $G$. We work in the context of order-unit spaces and consider orthogonal projections $P_\Lambda$ in $L^2(G)$ corresponding to finite subsets of irreducible representations $\Lambda \subseteq \widehat G$. We then prove that the sequence of truncated state spaces $\{ S(P_\Lambda C(G) P_\Lambda)\}_\Lambda$ Gromov-Hausdorff converges to the original state space $S(C(G))$, when these are equipped with a metric associated to a Lip-norm which in turn is induced by the action of $G$.