For which positive integers $k$ and $r$ are there involutions $g_{n,i} \in {\rm S}_n$ $(n \in \mathbb{N}, \ i = 1, \dots, k)$ such that the following hold?:
for any $n$, the $g_{n,i}$ $(i = 1, \dots, k)$ generate ${\rm S}_n$,
there is a uniform upper bound on the orders of the groups generated by all but one of the $k$ generators $g_{n,i}$ $(i = 1, \dots, k)$, valid for all $n \in \mathbb{N}$, and
there is a uniform upper bound on the orders of all words of length $\leq r$ in the $g_{n,i}$ $(i = 1, \dots, k)$, valid for all $n \in \mathbb{N}$.
Remark: The motivation for asking this question is that when trying to find an answer to this question, I came across a way to construct such permutations in the case $k = r = 6$, but only for $n$ congruent to $4$ or $15$ modulo $24$.
