Let $X,Y$ be metric space and suppose that $f:X\rightarrow Y$ is a *uniform embedding*; i.e.:
$$
\omega(d_X(x,z))\leq d_Y(f(x),f(z)) \leq \Omega(d_X(x,z)),
$$
where $\omega\leq \Omega$ are both strictly increasing continuous functions mapping $[0,\infty)$ to itself and which fix $0$.  

Is $f$ a quasisymmetry?  I.e.: does there exist a monotone function $\eta:[0,\infty)\rightarrow [0,\infty)$ satisfying
$$
\frac{d_Y(f(x),f(y))}{d_{Y}(f(x),f(z))} \leq \eta\left(\frac{d_X(x,y)}{d_X(x,z)}\right)
.
$$

**Note on Edit:** *I have reduced my previous question down to this more general one; since quasisymmetries preserve the doubling property.*