  A well-known theorem by J. Becker states that if a normalized univalent
function $f$ in the unit disk $\mathbb{D}$ can be embedded as the initial
element into a Loewner chain $(f_t)_{t\geqslant 0}$ such that the Herglotz
function $p$ in the Loewner -- Kufarev PDE $$\partial f_t(z)/\partial
f=zf'_t(z)p(z,t),\qquad z\in\mathbb{D},\quad\mathrm{a.e.}~t\ge0,$$ satisfies
$\big|(p(z,t)-1)/(p(z,t)+1)\big|\le k<1$, then $f$ admits a $k$-q.c.
(="$k$-quasiconformal") extension $F:\mathbb{C}\to\mathbb{C}$. The converse is
not true. However, a simple argument shows that if $f$ has a $q$-q.c. extension
with $q\in(0,1/6)$, then Becker's condition holds with $k:=6q$. In this paper
we address the following problem: find the largest $k_*\in(0,1]$ with the
property that for any $q\in(0,k_*)$ there exists $k_0(q)\in(0,1)$ such that
every normalized univalent function $f:\mathbb D\to\mathbb C$ with a $q$-q.c.
extension to $\mathbb C$ satisfies Becker's condition with $k:=k_0(q)$. We
prove that $k_*\ge1/3$.
