for a.e.$t\ge0$ and, moreover, generates the same function $f$. The latter can be verified using the change of variables $\tau:=Q(t)$, $\omega(\tau):=e^{i\Im Q(t)}w(t)$ that transforms the Loewner–Kufarev ODE  to ${\mathrm{d}}\omega/{\mathrm{d}}\tau=-\omega p_0(\omega,\tau)$.

Note that $L^{\!{\mathbb{D}}}_t:=H\circ L_t\circ H^{-1}$, where $H(\zeta):=(\zeta-1)/(\zeta+1)$, is an automorphism of ${\mathbb{D}}$ that sends $z_0(t):=H\big(p_t(0,t)\big)$ to $0$. Taking into account that by , $H\big(p({\mathbb{D}},t)\big)\subset k\overline{\mathbb{D}}$ for a.e.$t\ge0$, we see that $H\big(p_0({\mathbb{D}},t)\big)=L_t^{\!{\mathbb{D}}}\big(H\big(p({\mathbb{D}},t)\big)\big)$ is contained for a.e.$t\ge0$ in $\kappa\overline{\mathbb{D}}$, where $\kappa:=2k/(1+k^2)$. The conclusion of the proposition follows immediately.

One natural way to attack the above Problems \[PRO\_strong\] and \[PRO\_weak\] would be to propose several constructions of Loewner chains $(f_t)$ starting from an arbitrary given function ${f_0=f\in{\mathcal{S}}_k}$, with images $f_t({\mathbb{D}})$ being Jordan domains for all $t\ge0$, and try to find out whether the map $F:{\mathbb{C}}\to{\mathbb{C}}$ defined by  is quasiconformal for any of these constructions.

Here we examine two quite natural constructions and show that unfortunately, both fail in general. Fix some locally absolutely continuous function $\omega:[0,+\infty)\to\partial{\mathbb{D}}$ and let ${\rho:[0,+\infty)\to[1,+\infty)}$ be a strictly increasing continuous function with ${\rho(0)=1}$ and ${\lim_{t\to+\infty}\rho(t)=+\infty}$.

1ex [**Construction 1.**]{} Let $\Phi:{\mathbb{C}}\to{\mathbb{C}}$ be a $k$-q.c. map such that $f_0:=\Phi|_{\mathbb{D}}\in{\mathcal{S}}$. For ${t\ge0}$, denote by $f^\Phi_t$, the conformal map of ${\mathbb{D}}$ onto $\Phi\big(\rho(t){\mathbb{D}}\big)$ normalized by ${f^\Phi_t(0)=0}$, ${\omega(t)(f^\Phi_t)'(0)>0}$. For a suitable choice of the 