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 parametrization of $\Gamma$? What about uniqueness?

I would really appreciate any reference of previous work in this direction, as well as ideas and suggestions.