Suffix-language characterization conjecture for FKM universal cycles

Let $\textbf{T}(n,k)$ denote the set of strings of length $n$ over the paper's alphabet, and let $\textbf{S}\subseteq\textbf{T}(n,k)$ be closed under rotations. Given a string $\alpha\in\textbf{T}(n,k)$, an $\alpha$-suffix language is a set of strings such that, whenever $\beta=b_1\cdots b_n$ belongs to it and $\alpha=a_1\cdots a_n$, one has $b_m\leq a_m$ and $b_1\cdots b_{m-1}a_m\cdots a_n$ belongs to the set for every $1\leq m\leq n$. Suffix-language characterization conjecture. The FKM algorithm generates a universal cycle for $\textbf{S}$ if and only if the set of necklaces in $\textbf{S}$ is an $\alpha$-suffix language, where $\alpha$ is the lexicographically maximal necklace in $\textbf{S}$. This proposes a necessary-and-sufficient characterization of the rotation-closed languages for which the FKM algorithm produces a universal cycle.