Reduced smoothness conjecture for cascades in continuous families

Let $f:\mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ be a continuous family and let $U \subset \mathbb{R} \times \mathbb{R}^n$ be such that the periodic orbits in $U$ are contained in a bounded region. Assume that the boundary of $U$ contains only hyperbolic periodic orbits. For a fixed odd number $d$ and any positive integer $m$, let $IN_d$ and $OUT_d$ be, respectively, the generalized entry and exit orbits on the boundary of $U$ having period $p=2^m d$. Assume that the numbers of orbits in $OUT_d$ and $IN_d$ differ, and let $K$ be the smaller of these two numbers, allowing one but not both to be infinite. Reduced smoothness conjecture. There is a cascade through all but possibly $K$ of the orbits in the larger of $IN_d$ and $OUT_d$. This is intended to extend the abstract cascade theorem to non-generic perturbations, where controlling eigenvalues not involved in bifurcations is the main obstacle; the supplied text does not establish the claim.