The Brazilian Dream Problem for uniqueness of quasiconformal extensions

Let $E:{BL}_+(\mathbb{R})\to QC(\mathbb{H})$ be an extension satisfying properties (P1), (P2), (P3) and (P4). Here $\mathcal{E}_{a,\alpha}$ denotes the family of extensions parametrized by $a\in\mathbb{R}$ and $\alpha>0$.

The Brazilian Dream Problem. Then

$$E=\mathcal{E}_{a,\alpha}$$

for some $a\in\mathbb{R}$ and $\alpha>0$.

This is presented as a conjecture about the uniqueness of extensions satisfying the four stated properties; the authors report that they were unable to prove it and believe it to be true.