The extremal length conjecture for tournaments and the cyclic tournament

Let $[n]=\{1,\ldots,n\}$, let $\mathrm{Tour}_n$ denote the tournaments on $[n]$, and let $\ell(T,r)$ be the maximum word length of a rank-$r$ transformation generated by the arcs of $T$. Define the tournament $\pi_n$ on $[n]$ by

$$E(\pi_n):=\{(i,(i+1)\bmod n):i\in[n]\}\cup\{(i,j):j+1<i\}.$$

Write $\ell_{\max}^{\mathrm{Tour}}(n,r)$ and $\ell_{\max}^{\mathrm{Tour}}(n)$ for the corresponding maxima over tournaments and ranks.

The extremal length conjecture. For every $n\geq 3$, $r\in[n-1]$, and $T\in\mathrm{Tour}_n$,

$$\ell(T,r)\leq\ell(\pi_n,r)=\ell_{\max}^{\mathrm{Tour}}(n,r),$$

with equality if and only if $T\cong\pi_n$. Furthermore,

$$\ell(\pi_n)=\ell_{\max}^{\mathrm{Tour}}(n)=\frac{n^2+3n-6}{2},$$

which is achieved for $\alpha:=n\ (n-1)\ \dots\ 2\ n$.

The conjecture identifies $\pi_n$ as the unique extremal tournament for these transformation-semigroup word lengths. The paper notes that $\pi_n$ is known to have the minimum number of strong subtournaments among strong tournaments, but the stated extremal length claim itself is proposed as a conjecture.