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.