Robust fast growth for generic families of diffeomorphisms

Let $M$ be the manifold under consideration, let $r\in\{1,\ldots,\infty,\omega\}$, and write $\operatorname{Diff}^r(M)$ for the space of $C^r$ diffeomorphisms of $M$. A family $(f_a)_{a\in\mathbb R^k}$ is $C^r$-generic if it belongs to the relevant residual subset of the space of such families, and “the growth of the number of periodic points is fast” has the meaning used in the paper. Robust fast-growth conjecture. For every $r\in\{1,\ldots,\infty,\omega\}$, there exists an open set $U\subset\operatorname{Diff}^r(M)$ such that, for every $k\geq 0$ and every $C^r$-generic family $(f_a)_{a\in\mathbb R^k}$ with $f_a\in U$, the growth of the number of periodic points of $f_a$ is fast for every sufficiently small $a$. The $C^\infty$ case of the related problem remains open; the proposed statement is motivated by the preceding results on fast growth and by the author's expectation that this formulation has a negative answer in that case.