$F$ in Construction 1 generated by the Loewner chain $(f^\Phi_t)$ is not quasiconformal, whichever $\rho$ and $\theta$ we choose.

Let us now consider Construction 2 with the same function ${f\in{\mathcal{S}}_{1/\sqrt{2}}}$ as above. One remarkable property of $\Omega:=f({\mathbb{D}})$ is that $\{1/z\colon z\in{\overline{\mathbb{C}}}\setminus\overline{\Omega}\}=-\tfrac14\Omega.$ It follows that, up to rotation, $g(z)=-4/f(-1/z)$ for all $z\in{\mathbb{C}}\setminus\overline{\mathbb{D}}$. Suppose that for a suitable choice of the functions $\rho$ and $\omega$, the homeomorphism $F:{\mathbb{C}}\to{\mathbb{C}}$ defined with the help of the Loewner chain $(f^g_t)$ is $k$-quasiconformal for some ${k\in(0,1)}$. Then arguing as in [@Istvan proof of Theorem 2], we see that for all ${t\ge0}$ aside from some null-set $N$, ${\vphantom{f_t}\smash{\stackengine{-.075ex}{f_t}{\boldsymbol{\cdot\hskip.175em}}{O}{r}{F}{\useanchorwidth}{S}}}:=\partial f_t/\partial t$ and $f_t'$ exist a.e. on $\partial{\mathbb{D}}$, do not vanish, and $$\label{EQ_limit-property}
\frac{{\vphantom{f_t}\smash{\stackengine{-.075ex}{f_t}{\boldsymbol{\cdot\hskip.175em}}{O}{r}{F}{\useanchorwidth}{S}}}(e^{i\theta})}{e^{i\theta}f_t'(e^{i\theta})}\,\in\,U(k),$$ where $U(k)$ is defined in Theorem \[TH\_Becker\]. Moreover, by construction, $\partial f_t({\mathbb{D}})$ is $C^{\infty}$ when ${t>0}$. Hence, in fact, $f_t'$ extends smoothly to $\partial{\mathbb{D}}$ for all $t>0$; see e.g. [@Pommerenke:BB Chapter 3].

Taking into account that $\big|g^{-1}\big(f_t(e^{i\theta})\big)\big|=\rho(t)$ for all ${t\ge0}$ and all ${\theta\in[0,2\pi]}$, it follows that $\rho'(t)$ exists for any $t\in(0,+\infty)\setminus N$ and for the normal velocity of $\partial f_t({\mathbb{D}})$ we have $$|f'_t(e^{i\theta})|\,\Re\!\Big(\frac{{\vphantom{f_t}\smash{\stackengine{-.075ex}{f_t}{\boldsymbol{\cdot\hskip.175em}}{O}{r}{F}{\useanchorwidth}{S}}}(e^{i\theta})}{e^{i\theta}f'(e^{i\theta})}\Big)~=~\frac{\Re\big({\vphantom{f_t}\smash{\stackengine{-.075ex}{f_t}{\boldsymbol{\cdot\hskip.175em}}{O}{r}{F}{\useanchorwidth}{S}}}(e^{i\theta})\,\overline{e^{i\theta}f'(e^{i\theta})}\,\big)}{|f'_t(e^{i\theta})|}~=~\rho'(t)\,\big|g'\big(g^{-1}(f_t(e^{i\theta}))\big)\big|.$$ Together with  this implies that on the one hand, for any $t\in(0,+\infty)\setminus N$, $$\label{EQ_ravn}
\frac1K\le\left|\frac{\rho'(t)\,g'\big(g^{-1}(f_t(e^{i\theta}))}{f_t'(e^{i\theta})}\right|\le K:=\frac{1+k}{1-k}\quad \text{for all~$\theta\in[0,2\pi]$}.$$

On the other hand, 