Rigidity of flat labels modulo Möbius transformations

Let $f\in C_{\hat{\alpha},\hat{\eta}}$ be a flat label, where $C_{\hat{\alpha},\hat{\eta}}$ denotes the space of labels with the prescribed data $\hat{\alpha}$ and $\hat{\eta}$. A deformation through flat labels is a deformation remaining in this space. Flat-label rigidity conjecture. The only deformations of $f$ through flat labels correspond to Möbius transformations. This conjecture concerns the uniqueness of solutions to the disk-packing problems modulo Möbius transformations; the preceding discussion establishes Möbius invariance, while the claimed uniqueness is left open.