The follower-set growth conjecture for subshifts

Let $X$ be a subshift, meaning a closed, shift-invariant subset of $A^{\mathbb{Z}}$ for a finite set $A$. For each $n$, let $F_X(n)$ be the set of distinct follower sets of words of length $n$ in $X$, where a follower set consists of the one-sided infinite sequences that may follow the word in some point of $X$. Follower-set growth conjecture. If there exists an $n$ such that

$$|F_X(n)|\leq n,$$

then $X$ is sofic. It is known that a subshift is sofic exactly when it has finitely many distinct follower sets, and the paper verifies the conjecture for $n\leq 3$, proves it for a large class of coded subshifts, and establishes the stronger sufficient condition $|F_X(n)|\leq \log_2(n+1)$.