The periodic-attractor bound for perturbations of one-dimensional maps

Let $f:{\cal N} \to {\cal N}$ be a $C^3$ map with non-flat critical points. Suppose that $f$ does not have neutral periodic orbits.

Periodic-attractor bound. There exists a neighbourhood ${\cal F} \subset C^3({\cal N})$ of $f$ such that for any $g\in {\cal F}$ the number of periodic attractors of $g$ is bounded by the number of attractors of $f$ plus the number of critical points of $f$ counted with their multiplicities.

This conjecture asserts the second part of the theorem discussed in the source for maps with non-flat critical points, including perturbations in which critical points may disappear or lose their degree. The surrounding examples show why the bound must account for critical points with their multiplicities; the status of the conjecture is not determined by the supplied text.