Alphabet-independence conjecture for the threshold avoiding powers and anti-powers

Let $N_\alpha(k,k)$ denote the quantity defined in the paper as the relevant threshold for words over an alphabet of size $\alpha$, avoiding $k$-powers and $k$-anti-powers. Alphabet-independence conjecture. The quantity $N_\alpha(k,k)$ is independent of $\alpha$. The few known values suggest this possible structural property of the set of words avoiding $k$-powers and $k$-anti-powers that achieve the length $N_\alpha(k,k)$ for arbitrary alphabet sizes; the conjecture is presented as an open direction.