Set theory – Existence of a bijective function that maps the subset tensor product of a selective ultrafilter in the ultrafilter

In the answer to this question, Andreas Blass had shown that for any selective ultrafilter $ scr {U} $ sure $ omega $ and for any free subfilter $ scr {F} subset {U} $ does not exist bijection $ varphi: omega ^ 2 to omega $ such as $ varphi ( scr {F} otimes scr {F}) subset scr {U} $. I'm trying to weaken the conditions.

Question: Is there a pair of subassemblies $ scr {A}, scr {B} $ selective ultrafilter $ scr {U} $ sure $ omega $ and a bijection $ varphi: omega ^ 2 to omega $ with the following properties:

  1. $ scr {A} $ and $ scr {B} $ have ownership of the finished intersections and $ cap scr {A} = cap scr {B} = varnothing $
  2. $ varphi ( scr {A} otimes scr {B}) subset scr {U} $ ?