Suppose $\mathscr{A}$ is a complete Boolean algebra (assume it is c.c.c. if you wish). Is there any, say, canonical embedding $\mathscr{A}\subseteq \mathscr{B}$ into a complete Boolean algebra which is weakly countably distributive ($(\omega, \omega)$-distributive)? Note that I do not put any extra assumptions on $\mathscr{B}$ but I would like to have it as small as possible.

One way to produce such an algebra is taking

$$\mathscr{B}=\mbox{Clop}(\mbox{Gelfand_spectrum}(C(\mbox{Stone}(\mathscr{A}))^{**})$$

provided the Gelfand spectrum of $C(\mbox{Stone}(\mathscr{A}))^{**}$ is c.c.c. This is however, in a certain vague sense, much larger that $\mathscr{A}$.