Density of jointly periodic points for surjective one-dimensional cellular automata

Let $\Sigma_N$ be the space of doubly infinite sequences over an alphabet of $N$ symbols, and let $f:\Sigma_N\to\Sigma_N$ be a surjective one-dimensional cellular automaton. Let $S_N$ be the shift map, and call a point jointly periodic if it is periodic under both $f$ and $S_N$. A subset of $\Sigma_N$ is dense if every finite central word occurring in a point of $\Sigma_N$ also occurs in some point of the subset. Density conjecture. For every surjective one-dimensional cellular automaton, the jointly periodic points are dense. This is a known open question, including in higher dimensions. It is known for closing maps and for surjective maps with a point of equicontinuity; computational evidence shows, in particular, that every span-$4$ surjective one-dimensional cellular automaton on two symbols has jointly periodic points that are at least $13$-dense.