Let $K$ be a real closed field of transcendence degree 1 over $\mathbb{R}$. It is not difficult to see that $K$ has the following "minimality property": Whenever $L$ is a real closed field that realizes all cuts of the rational numbers, then $K$ embeds into $L$.

**Question:** Is $K$ up to isomorphism uniquely determined by this minimality property?

**Terminology:**
By a *cut* of $\mathbb{Q}$ I mean a pair $(L,R)$ of subsets of $\mathbb{Q}$
with the property that $l < r$ for all $l\in L$, $r\in R$ and such that $R\cup L=\mathbb{Q}$

The question asks whether $K$ can be identified parallel to the way one can define $\mathbb{R}$:
The real field $\mathbb{R}$ is up to isomorphism the unique real closed field that embeds into any real closed field which realizes every *non-principal* cut of $\mathbb{Q}$; a cut $(L,R)$ is non-principal if $L$ does not have a supremum in $\mathbb{Q}\cup \{\pm\infty\}$