MA$_{\omega_1}(S)$ is the statement that $S$ is a Suslin tree and for any c.c.c. poset that preserves $S$ to be Suslin, for any collection of $\omega_1$ many dense subsets, there is a filter meeting them all. MA$_{\omega_1}(S)[S]$ holds if the universe is a forcing extension by $S$ over a model of MA$_{\omega_1}(S)$. $\mathcal{K}_2$ is the assertion that every c.c.c. partition of $[\omega_1]^2$ has an uncountable 0-homogeneous subset. And a partition $[\omega_1]^2=K_0\cup K_1$ is a \emph{c.c.c. partition} if every uncountable family of finite 0-homogeneous sets contains two members whose union is also 0-homogeneous.

Larson and Todorcevic asked if MA$_{\omega_1}(S)[S]$ implies $\mathcal{K}_2$. A positive answer will distinguish two closely related properties -- $\mathcal{K}_2$ and MA$_{\omega_1}$ -- since MA$_{\omega_1}$ fails in models of MA$_{\omega_1}(S)[S]$. However,   we answer this question negatively. This is a joint work with Liuzhen Wu.