Conjecture on free abelian subgroups of cactus groups

Let $J_n$ be the cactus group on $n$ strands, and let $k$ be a nonnegative integer. A free abelian subgroup of rank $k$ is a subgroup isomorphic to $\mathbb{Z}^k$. Free abelian subgroup conjecture. The cactus group $J_n$ contains a free abelian subgroup of rank $k$ if and only if

$$k\leq\left\lfloor\frac{n}{3}\right\rfloor.$$

A free abelian subgroup of rank $\lfloor n/3\rfloor$ is already known to exist, so the conjecture asserts that this construction has maximal possible rank and determines the algebraic dimension of $J_n$.