Immediate renormalization of cubic complex polynomials with empty rational lamination

A cubic polynomial $P$ with a non-repelling fixed point $b$ is said to be immediately renormalizable if there exists a (connected) QL 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.