Mori's exact constant conjecture for planar quasiconformal maps

Let $QC_K$ be the family of all $K$-quasiconformal self-maps of the unit disk $\mathbf{B}^2$ that fix the origin, and let $M(2,K)$ be the least constant such that

$$|f(x)-f(y)|\leq M(2,K)|x-y|^{K^{-1}},$$

for every $f\in QC_K$ and $x,y\in\mathbf{B}^2$. Mori's conjecture.

$$M(2,K)=16^{1-1/K}.$$

The conjecture refines Mori's bound $M(2,K)\leq 16$ by incorporating the exact value at $K=1$; it was still open in 2009.