  A metric space $X$ is quasisymmetrically co-Hopfian if every quasisymmetric
embedding of $X$ into itself is onto. We construct the first examples of metric
spaces homeomorphic to the universal Menger curve and higher dimensional
Sierpi\'nski spaces, which are quasisymmetrically co-Hopfian. We also show that
the collection of quasisymmetric equivalence classes of spaces homeomorphic to
the Menger curve is uncountable. These results answer a problem and generalize
results of Merenkov from \cite{Mer:coHopf}.
