Self-similarity conjecture for open subgroups of norm-one groups

Let $K$ be a finite extension of $\mathbb{Q}_p$ with ramification index $e$ and residue degree $d$, let $D$ be a central division algebra over $K$ of degree $n$, let $\Delta$ be a maximal order in $D$, and write $SL_1(D)$ for the group of elements of reduced norm one. For $l\in\mathbb{N}$, set

$$SL_1^l(D)=SL_1(D)\cap(1+\mathfrak{p}^l),$$

where $\mathfrak{p}$ is the unique maximal two-sided ideal of $\Delta$.

Self-similarity conjecture. Assume that $n\geq 2$ and $p\geq d(n^2-1)$. Then no open subgroup of $SL_1^{ne}(D)$ is self-similar.

The conjecture generalizes the cited conjecture for self-similarity of norm-one groups and is supported by the theorem in the paper, which proves the corresponding non-self-similarity statement for $SL_1^{mne}(D)$ under the stated hypotheses. Its full assertion for every open subgroup remains unresolved in the supplied text.