The maximum-row-size conjecture for prefix-reversal generating triples

For integers $n$ and $k$ with $n>k+1$, define

$$F_k(n)=\{m:k<m<n,\ \langle r_n,r_m,r_k\rangle=\operatorname{Sym}_n\}.$$

Maximum-row-size conjecture.

$$\max_{n>k+1}|F_k(n)|=\begin{cases} k+1&\text{if }k\text{ is even},\\ \frac{k+1}{2}&\text{if }k\text{ is odd}. \end{cases}$$

The conjecture is motivated by computational experiments and the displayed generating-triple patterns; the source gives no proof or resolution.