A topological space $X$ has the c.c.c. if every family of pairwise disjoint non-empty open subsets of $X$ is countable. Consider the following statement:
$(P)$ The product of two c.c.c. spaces is c.c.c.
It is known that that Martin's Axiom $\mathrm{MA}(\omega_1)$ implies $(P)$. A reference is Kunen's Set theory, Chapter II.
Question. Does the converse also hold? If not, how can we force $(P) \wedge \neg \mathrm{MA}(\omega_1)$?
By the way, $(P)$ is equivalent to the statement that c.c.c. spaces are closed under arbitrary products, not just binary products.
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).