  A cubic polynomial $P$ with a non-repelling fixed point $b$ is said to be
\emph{immediately renormalizable} if there exists a (connected) quadratic-like
invariant filled Julia set $K^*$ such that $b\in K^*$. In that case exactly one
critical point of $P$ does not belong to $K^*$. We show that if, in addition,
the Julia set of $P$ has no (pre)periodic cutpoints then this critical point is
recurrent.
