Existence of a low-growth surjective cellular automaton

Let $\Sigma_N$ be the space of doubly infinite sequences over an alphabet of $N$ symbols, let $S_N$ be its shift map, and let $f:\Sigma_N\to\Sigma_N$ be a surjective one-dimensional cellular automaton. For each $k$, let $P_k(S_N)$ be the set of points fixed by $S_N^k$, let $P=|\operatorname{Per}(f|P_k(S_N))|$ be the number of points in $P_k(S_N)$ that are also periodic under $f$, and define

$$\nu_k(f,S_N)=P^{1/k},\qquad \nu(f,S_N)=\limsup_k\nu_k(f,S_N).$$

Low-growth existence conjecture. There exists $N>1$ and a surjective cellular automaton $f$ on $N$ symbols such that $\nu(f,S_N)<N$. The surrounding discussion poses related quantitative questions about lower bounds for $\nu(f,S_N)$, while this existence claim is presented without a supplied resolution status.