Strengthened equivalence-class bound for increasing and decreasing pattern replacements

Let $S_n$ be the set of permutations of $\\{1,2,\ldots,n\\\\}$, and let two permutations be equivalent when one can be obtained from the other by replacing an occurrence of the pattern $12 \cdots k$ with $k \cdots 21$, or vice versa. Strengthened equivalence-class conjecture. Let $k \ge 3$. If $n \ge k^2-2k+3$, then there are only one or two equivalence classes under the equivalence $\\{12 \cdots k, k \cdots 21\\}$. Furthermore, if $k$ is even, it suffices to assume $n \ge k^2-2k+2$. This strengthens the preceding theorem, which gives the same conclusion only for $n \ge 3k^2-4k+3$; the conjecture had been experimentally verified for $k \le 4$ in the source.