For a set $X$ we endow the set $\omega^X$ of all functions from $X$ to $\omega$ with the natural partial order $\le$ defined by $f\le g$ iff $f(x)\le g(x)$ for all $x\in X$.

A function $f:\omega^\omega\to\omega^X$ is called monotone if for any $\alpha\le\beta$ in $\omega^\omega$ we have $f(\alpha)\le f(\beta)$ in $\omega^X$.

Question. Is there a monotone function $f:\omega^\omega\to\omega^{\omega_1}$ such that for every countable set $A\subset\omega_1$ and every $\alpha\in\omega^A$ there exists $\beta\in\omega^\omega$ such that $\alpha\le f(\beta){\restriction}_A$?

Remark. By Proposition 2.4.1(2) in this paper, for every monotone function $f:\omega^\omega\to\omega^{\omega_1}$ there exists $\alpha\in\omega^{\omega_1}$ such that for every $\beta\in\omega^\omega$ we have $\alpha\not\le f(\beta)$.

PS. I learned this question from Jerzy Kąkol who arrived to it studying $\mathfrak G$-bases in locally convex spaces.