Question.Does a quasiconformal map exist between a subset of $\mathbb{C}$ (such as a unit disc or rectangle) and a polytope?

Here, I take a polytope to be a two-dimensional surface that could be embedded in $\mathbb{R}^3$ or some other three-dimensional space, possibly $\mathbb{H}^3$.

In particular, this question interested me as it would offer a method to parameterize a permutohedron with the complex plane. The premise of this question was inspired by Dantzig's method, a numerical optimization technique in linear programming that can be realized as a path along the edge of a simplex.

By segmenting $\mathbb{R}^3$ into a series of planes and applying a Schwarz-Christoffel transform on each, I was able to form an injective map from the unit disc to polytope in three dimensions; however, I do not believe it is quasiconformal. As a secondary question, does anyone know if any such mapping could preserve the holomorphicity of a function on $\mathbb{C}$?

**Edit (Clarification):** Could it be possible to make a quasiconformal or conformal map from a disk in $\mathbb{C}$ to the following polytope?

Thank you.