This is problem 15.3 in Arnie Miller's problem list:

(Juhasz) Suppose there exists $\langle A_{\alpha} : \alpha \in L \rangle$, where $L$ is the set of limit ordinals below $\omega_1$ and for each $\alpha \in L$, $A_{\alpha}$ is an unbounded subset of $\alpha$, satisfying: For every unbounded $A \subseteq \omega_1$, there exists $\alpha \in L$, $A_{\alpha} \subseteq A$. Must there exists a Suslin tree?

What is the current status of this problem? Thanks!