Consider the maximal unramified extension $\mathbb{Q}_p^{nr}$ of the field $\mathbb{Q}_p$ of $p$-adic numbers. Its residue field is equal to $\overline{\mathbb{F}_p}$. Consider now some field $K$ satisfying $\mathbb{F}_p \subseteq K \subseteq \overline{\mathbb{F}_p}$ and consider the corresponding unramified extension $F$ of $\mathbb{Q}_p$, which is of course contained in $\mathbb{Q}_p^{nr}$.
Under what conditions on $K$ is the field $F$ dense in $\mathbb{Q}_p^{nr}$ for the topology given by the valuation?