Non-word-representability conjecture for simplified de Bruijn graphs

Let $S(n,k)$ denote the simple graph obtained from the de Bruijn graph by removing orientations and loops and replacing multiple edges between a pair of vertices by a single edge. A graph is word-representable if there is a word over its vertex set in which two letters alternate if and only if they are adjacent.

Non-word-representability conjecture. All simplified de Bruijn graphs $S(n,k)$ are non-word-representable for $n\geq 4$ and $k\geq 3$.

The paper proves that binary simplified de Bruijn graphs $S(n,2)$ are word-representable for every $n\geq 1$, while $S(2,k)$ and $S(3,k)$ are non-word-representable for $k\geq 3$. The conjecture extends the latter non-word-representability results to all alphabet sizes $n\geq 4$ when $k\geq 3$.