The conjectures on relation numbers and congruence structure of $\Delta_{r/p}$

Let $p$ be prime and $r\in\mathbb{Z}$ satisfy $\gcd(r,p)=1$, with $r/p$ admissible as defined in the source. Let $\Delta_{r/p}$ be generated by the two parabolic matrices $A$ and $Q_{r/p}$, let $U_p$ denote the diagonal matrix used in the source, let $\sigma_p(r)$ be the multiplicative order of $p$ modulo $r$, and let $J_2(r)$ be the index quantity used there. The conjectures for admissible parameters. For every admissible $r/p$, all seven assertions listed in the source hold: $r/p$ is a relation number; $\Delta_{r/p}$ has finite index and is a congruence subgroup; $\Delta_{r/p}=\overline{\Gamma}_1^{(p)}(r)$ with index $J_2(r)$; $U_p^{\sigma_p(r)}\in\Delta_{r/p}$; $r/p$ is a strong relation number; $\Delta_{r/p}=\Delta_{r/p^k}$ for every $k\geq1$; and $\operatorname{SL}_2\left(\mathbb{Z}[\frac1p]\right)=\Delta_{r/p}\cdot\operatorname{SL}_2(\mathbb{Z})$. These are collected as conjectures at the end of the paper; their general status is unresolved.