Simplicity conjecture for the subgroup of the Burger–Mozes group

Let $\Lambda$ be the finitely presented $(6,6)$-group defined by the generators and relators in the preceding construction, and let $\Lambda_0<\Lambda$ be the subgroup under consideration. Simplicity conjecture. The subgroup $\Lambda_0<\Lambda$ is simple. The preceding proposition establishes that every non-trivial normal subgroup of $\Lambda$ has finite index; computational experiments on finite-index subgroups motivate the stronger simplicity claim for $\Lambda_0$, which remains unresolved in the supplied text.