Martin's Axiom and products of c.c.c. spaces
Solution 1:
I believe this question is open. See for example the paper 'Chain Conditions in Maximal Models' by Larson and Todorcevic where this and other related questions are discussed. In particular, this question is Question 2 in Section 10 (what you call $(P)$ they call $\mathcal{C}^2$, with the definition given in Section 4).