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 $\varphi(\scr{F}\otimes\scr{F})\subset\scr{U}$, because $\scr{U}$ is P-point and P-point is exactly an ultrafilter which isomorphic image in $\omega^2$ can not contain $\scr{N}\otimes\scr{N}$ for Fréchet filter $\scr{N}$. 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 $\cap\scr{A}=\cap\scr{B}=\varnothing$
- $\varphi(\scr{A}\otimes\scr{B})\subset\scr{U}$ ?