The self-similarity conjecture for open subgroups of norm-one quaternion groups

Let $p$ be an odd prime, and let $\Delta_p$ denote the relevant quaternion division algebra over $\mathbb{Q}_p$. The groups $SL_1^1(\Delta_p)$ and $SL_1^2(\Delta_p)$ are the principal congruence subgroups of its norm-one group.

Self-similarity conjecture. For all primes $p\geqslant 5$, no open subgroup of $SL_1^1(\Delta_p)$ is self-similar. For all primes $p\geqslant 3$, no open subgroup of $SL_1^2(\Delta_p)$ is self-similar.

This conjecture complements the paper’s self-similarity results for the $SL_2$-type groups and asserts a dichotomy among open subgroups of the two three-dimensional nonsolvable families. The supplied text does not indicate whether it has been resolved.