Let P be the statement: Every subset of plane belongs to the sigma algebra generated by $\{A \times B : A, B \subseteq \mathbb{R}\}$.

Let Q be the statement: Every continuum-sized family of subsets of $\mathbb{R}$ is contained in a countably generated sigma-algebra.

Both statements are independent of ZFC and P implies Q. Does Q imply P?

This stems from the following question. As noted by Gro-Tsen, it was also asked in Bing, Bledsoe & Mauldin, "Sets Generated by Rectangles", Pacific J. Math. 51 (1974) 27–36.

Arnie Miller has a related result: Letting $\kappa = \aleph_{\omega_1}$, it is consistent that $\mathfrak{c} = \kappa^+$, every $\kappa$-sized family of sets of reals is contained in a countably generated sigma algebra and there is a subset of $\kappa \times \mathfrak{c}$ which is not in the sigma-algebra of abstract rectangles. See Theorem 5.12 here or Theorem 55 in his thesis.