Let P be the statement: Every ccc partial order has $\omega_1$-precaliber; i.e., every uncountable subset $X$ of a ccc partial order $P$ has an uncountable subset $Y$ such that for every finite subset $F$ of $Y$, there is a member in $P$ below every member of $F$.

Let Q be the statement: Product of ccc partial orders is ccc.

It is known that $MA(\omega_1)$ (Martin's axiom at $\omega_1$) implies $P$ and that $P$ implies $Q$. Do these implications reverse?