Conjecture that the principal arithmetic subgroup has no non-trivial symmetries

Let $N(r)$ denote the order of the outer automorphism group of the principal arithmetic subgroup $\Lambda_0^n$, in the setting of the arithmetic quotients considered in the paper. Symmetry conjecture. For all $r\geq 2$, $N(r)=1$, and therefore

$$|\chi(O_{\min}^n)|=\frac{\lambda(r)}{4^{r-1}}\prod_{i=1}^{r}|\zeta_{{\mathbb Q}[\sqrt5]}(1-2i)|.$$

The conjecture asserts that the principal arithmetic subgroup has no non-trivial outer symmetries; the source notes that this can be checked in small dimensions where the group is reflective, but does not provide a proof in higher dimensions. The displayed formula gives the resulting expression for the Euler characteristic of the minimal orbifold.