The mod-2 persistence conjecture for prefix-reversal generating triples

Let $n,m,k$ be integers with $2\leqslant k<m<n$. Mod-2 persistence conjecture. For any $n\equiv0,1\pmod{4}$, if

$$\langle r_n,r_m,r_k\rangle=\operatorname{Sym}_n,$$

then

$$\langle r_{n+2},r_{m+2},r_{k+2}\rangle=\operatorname{Sym}_{n+2}.$$

This was proposed from computational experiments for $n\leqslant100$; no proof or resolution is given.