The order-three conjecture for cactus group matrices of $\mathfrak{sl}_3$

Let $\lambda=l_1\omega_1+l_2\omega_2\in P^+$, and let $N^{i;\lambda}$, for $i\in\{1,2\}$, be the matrices of the cactus-group generators $\sigma^i$ with respect to the basis $\mathbf B_\lambda$ of the irreducible module $V_\lambda$. Order-three conjecture. For every such $\lambda$,

$$(N^{1;\lambda}N^{2;\lambda})^3=1.$$

The relation asserts the expected order-three relation for the product of the two generators in the $\mathfrak{sl}_3$ cactus-group action. The statement was verified computationally using Mathematica for all $l_1,l_2\in\mathbb Z_{\ge 0}$ satisfying $l_1+l_2\le 14$, but the supplied source does not establish it in general.