In classical conformal welding theory, we start with a homeomorphism $h$ of the unit circle and try to find a Jordan domain $D$ together with two conformal isomorphisms $f_1 \colon \mathbb D \to D$ and $f_2 \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \overline{D}$. such that $h|_{S^1} = f_1^{-1} \circ f_2$. The existence of these two maps are guaranteed if $h$ is a quasisymmetric homeomorphism, and the uniqueness follows from uniqueness of complex dilatation.
My question is: Let $h$ be a homeomorphism of the unit circle, and let $J(z) = z+1/z$. Under what conditions can we find a rectifiable curve $\Gamma$ of length $4$ together with a conformal isomorphism $f \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma$ such that $f|_{S^1} = \gamma \circ J \circ h$, where $\gamma \colon [-2,2] \to \mathbb C$ is an arc-length parameterization of $\Gamma$? What about uniqueness?
I would really appreciate any reference of previous work in this direction, as well as ideas and suggestions.
Some motivations for this question:
Right now, I am trying to classify all smooth curves (and eventually rectifiable curves or Jordan arcs) $\Gamma$ with a certain property that reduces to $$\gamma'(t) = \Phi_1^{-1}(\gamma(t)) \Phi_2^{-1}(\gamma(t)).$$ To explain the symbols, $\gamma$ is an arc-length parameterization of $\Gamma$, and $\Phi_1^{-1}(z), \Phi_2^{-1}(z)$ represent the two pre-images of $z \in \Gamma$ under the conformal isomorphism $\Phi \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma$. These pre-images are distinct if $z$ is not one of the endpoints of $\Gamma$.
I know curves of constant curvature satisfy this property, and I have some reasons to believe that they are the only ones. I would hope the "uniqueness" of conformal welding of curves, in some appropriate sense, would help me with the classification.
