Conjectured intersection property for the constructed generators

Let $\alpha$, $\beta$, and $\gamma$ be the three generators constructed from the elements in the preceding construction, and let $\langle\alpha,\beta,\gamma\rangle$ denote the subgroup they generate. The intersection-property conjecture asserts that the set $\{\alpha,\beta,\gamma\}$ satisfies

$$\langle\alpha,\beta\rangle\cap\langle\beta,\gamma\rangle=\langle\beta\rangle.$$

This property is the remaining group-theoretic condition needed for the constructed generators to define the desired chiral polytope. The paper states the conjecture without computational experiments, and its status is therefore open.