Conjecture on maximal finite subgroups of cactus groups

Let $J_n$ be the cactus group on $n$ strands. A finite group embeds into $J_n$ precisely when it embeds into an iterated permutational wreath product of copies of $\mathbb{Z}/2\mathbb{Z}$ of the form

$$(\cdots (\mathbb{Z}/2\mathbb{Z} \wr_{r_1} \mathbb{Z}/2\mathbb{Z}) \wr_{r_2} \cdots ) \wr_{r_s} \mathbb{Z}/2\mathbb{Z},$$

where $r_1,\ldots,r_s$ satisfy $r_1\cdots r_s\leq n/2$. Here, for a group $A$ and $r\geq 1$, $A\wr_r\mathbb{Z}/2\mathbb{Z}$ denotes $A^r\rtimes\mathbb{Z}/2\mathbb{Z}$, with the right factor reversing the $r$ factors. Maximal finite subgroup conjecture. A finite group embeds into the cactus group $J_n$ if and only if it embeds into

$$(\cdots (\mathbb{Z}/2\mathbb{Z} \wr_{r_1} \mathbb{Z}/2\mathbb{Z}) \wr_{r_2} \cdots ) \wr_{r_s} \mathbb{Z}/2\mathbb{Z}$$

for some $r_1,\ldots,r_s$ satisfying $r_1\cdots r_s\leq n/2$. This would give an explicit algebraic description of finite subgroups, beyond the fact that they are $2$-groups, using the median geometry of cactus groups.