# Dense subfilter of selective ultrafilter

Given selective ultrafilter $$\mathcal{U}$$ on $$\omega$$ and dense filter $$\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$$, where $$\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$$ if the limit exists. Let $$\mathcal{F}=\mathcal{F_1}\cap\mathcal{U}$$.

Question: Does there exist a family $$\{A_i\subset\omega\}_{i<\omega},~A_i=\{a_{ik}\}_{k<\omega}$$ of pairwise disjoint subsets such that for any $$B\in\mathcal{F}$$ we have: $$\{a_{ik}~|~i,k\in B\}\in\mathcal{U}$$

Remark: The question is equivalent formulation of this one which still has no answer.