In the answer on this question Andreas Blass had shown that for any selective ultrafilter $ \scr{U}$ on $ \omega$ and for any free subfilter $ \scr{F}\subset{U}$ doesn’t exist bijection $ \varphi:\omega^2\to\omega$ such that $ \varphi(\scr{F}\otimes\scr{F})\subset\scr{U}$ . Thus I am trying to weaken the conditions.

**Question:** Does there exist a pair of subsets $ \scr{A},\scr{B}$ of selective ultrafilter $ \scr{U}$ on $ \omega$ and a bijection $ \varphi:\omega^2\to\omega$ with following properties:

- $ \scr{A}$ and $ \scr{B}$ have finite intersections property and $ \cap\scr{A}=\cap\scr{B}=\varnothing$
- $ \varphi(\scr{A}\otimes\scr{B})\subset\scr{U}$ ?