  The solution of the Chern-Simons-Higgs model in Lorenz gauge with data for
the potential in $H^{s-1/2}$ and for the Higgs field in $H^s \times H^{s-1}$ is
shown to be unique in the natural space $C([0,T];H^{s-1/2} \times H^s \times
H^{s-1})$ for $s \ge 1$, where $s=1$ corresponds to finite energy. Huh and Oh
recently proved local well-posedness for $s > 3/4$, but uniqueness was obtained
only in a proper subspace $Y^s$ of Bourgain type. We prove that any solution in
$C([0,T];H^{1/2} \times H^1 \times L^2)$ must in fact belong to the space
$Y^{3/4+\epsilon}$, hence it is the unique solution obtained by Huh and Oh.
