Forbidden-substring conjecture for greedy universal cycles

Let $\textbf{T}(n,k)$ denote the set of strings of length $n$ over the alphabet of symbols used in the paper. Let $\gamma\in\textbf{T}(m,k)$ be a string containing a symbol $a\leq k-2$, and let $\textbf{S}\subseteq\textbf{T}(n,k)$, for some $n\geq m$, be the set of strings not containing $\gamma$ as a cyclic substring. Forbidden-substring conjecture. The greedy algorithm starting from $k^n$ generates a universal cycle for $\textbf{S}$. This would complete the known partial classification of forbidden substrings for which the greedy algorithm generates a universal cycle; the claim concerns the remaining case in which the forbidden string contains a symbol at most $k-2$.