Let $F$ be a field. Does the category $C_F$ of local rings with residue field isomorphic to $F$ have an initial object?

This is, for instance, true if $F=\mathbb{F}_{p}$ for some prime $p$: If $R$ is a local ring with residue field $\mathbb{F}_{p}$, then any $x\in\mathbb{Z}\setminus(p)$ must map to something invertible under the morphism $\mathbb{Z}\longrightarrow R$. Hence that morphism factors as $\mathbb{Z}\longrightarrow\mathbb{Z}_{(p)}\longrightarrow R$; thus $\mathbb{Z}_{(p)}$ is the initial object.

But what happens in the more general case? I guess it should be true at least if $F$ is of finite type over $\mathbb{Z}$, but I have no idea how to prove it.

(EDIT - To avoid any confusion: I am talking about an initial object in the category of local rings $R$ with a fixed surjection $R\longrightarrow F$.)

Let $F$ be a field. Does the category $C_F$ of local rings $R$ equipped with a surjective morphism $R\longrightarrow F$ have an initial object?

This is, for instance, true if $F=\mathbb{F}_{p}$ for some prime $p$: If $R$ is a local ring with residue field $\mathbb{F}_{p}$, then any $x\in\mathbb{Z}\setminus(p)$ must map to something invertible under the morphism $\mathbb{Z}\longrightarrow R$. Hence that morphism factors as $\mathbb{Z}\longrightarrow\mathbb{Z}_{(p)}\longrightarrow R$; thus $\mathbb{Z}_{(p)}$ is the initial object.

But what happens in the more general case? I guess it should be true at least if $F$ is of finite type over either $\mathbb{Q}$ or $\mathbb{F}_{p}$ (where $p$ is a prime), but I have no idea how to prove it.