  We construct a model in which MA$_{\omega_1}$(S)[S] holds and $\mathcal{K}_2$
fails. This shows that MA$_{\omega_1}$(S)[S] does not imply $\mathcal{K}_2$ and
answers an old question of Larson and Todorcevic in [3]. We also investigate
different strong colorings in models of MA$_{\omega_1}$(S)[S].
