$\delta$-lifting and $1$-dimensional analytic fields

Let $k$ be an algebraically closed complete non-Archimedean field, and let $K$ be a finitely generated field extension over $k$ with transcendence degree $1$. Equip $K$ a non-Archimedean norm extending the one on $k$, and let $\mathcal{K}$ denote the completion of $K$. We will prove that the valuation ring $\mathcal{K}^+$ admits a flat $\delta$-lifting over $\mathbb{A}_{\mathrm{inf}}(k^+)$ if and only if $\mathcal{K}$ is not of type 4.