The minimum-length conjecture for odd tournaments and circulant tournaments

Let $[n]=\{1,\ldots,n\}$, let $\mathrm{Tour}_n$ denote the tournaments on $[n]$, and let $\kappa_n$ be the circulant tournament. Write $\ell(T,r)$ for the maximum word length of a rank-$r$ transformation generated by the arcs of $T$, and let $\ell_{\min}^{\mathrm{Tour}}(n,r)$ denote the minimum of this quantity over tournaments in $\mathrm{Tour}_n$.

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

$$\ell_{\min}^{\mathrm{Tour}}(n,r)=\ell(\kappa_n,r).$$

Furthermore,

$$\ell_{\min}^{\mathrm{Tour}}(n,2)=n+1$$

and

$$\ell_{\min}^{\mathrm{Tour}}(n,r)=n+r$$

for all $3\leq r\leq\frac{n+1}{2}$.

This conjecture proposes that the circulant tournament gives the minimum possible rank-wise length for odd orders, and specifies the minimum explicitly for ranks up to $(n+1)/2$. The source gives no resolution of the claim.