If $\mathcal{U}$ is a selective (Ramsey) ultrafilter on $\omega$ and $\mathcal{B}$ is its base, then for every sequence $A_0\supsetneq A_1\supsetneq A_2\supsetneq\ldots$ in $\mathcal{B}$ there exists $A\in\mathcal{B}$ such that $|A\setminus A_n|\le n+1$. I'm interested in the following generalization of this notion.
Let $\mathcal{B}$ be a base of a non-principal ultrafilter $\mathcal{U}$ and let $\mathcal{F}\subset[\omega]^\omega$. Let us say that $\mathcal{F}$ is selecting for $\mathcal{B}$, if for every sequence $A_0\supsetneq A_1\supsetneq A_2\supsetneq\ldots$ in $\mathcal{B}$ there exists $A\in\mathcal{F}$ such that $|A\setminus A_n|\le n+1$. (So, every base of a selective ultrafilter is selecting for itself). Define the following cardinal numbers:
$\mathfrak{sel}(\mathcal{U})=\min\{|\mathcal{F}|\colon\ \mathcal{F}\subset[\omega]^\omega\text{ is selecting for some base of }\mathcal{U}\}$,
$\mathfrak{sel}=\min\{\mathfrak{sel}(\mathcal{U})\colon\ \mathcal{U}\text{ is a non-principal ultrafilter}\}$.
Question 1: Does $\mathfrak{u}=\mathfrak{sel}$ hold in ZFC?
Question 2: If no, is it consistent that $\mathfrak{u}>\mathfrak{sel}$?
If this topic has been studied, then please give me some references. Thank you!
