Type $D_m$ cactus-generator conjecture for the crystal $B(n\omega_1)$

Let $\mathfrak g$ be of type $D_m$, let $I$ be its index set, and consider the crystal $B(n\omega_1)$ for $n\in\mathbb N$. The cactus group acts on this crystal, with cactus generators indexed by subsets of $I$.

Type $D_m$ cactus-generator conjecture. The action of the cactus group on $B(n\omega_1)$ is generated by the actions of cactus generators corresponding to one-element and two-element subsets of $I$.

The conjecture is based on computational evidence and proposes a small generating collection for the cactus-group action in type $D_m$. It is a specialization of the broader relationship between cactus actions and toggle groups described in the paper.