Centrality conjecture for the Sigma-congruence kernel

Let $\Sigma=\{q,\infty\}$, where $q$ is a prime as in the preceding construction, and let $B^{(1)}$ be the norm-one subgroup of the quaternion algebra $B$. Write $C(\Sigma)$ for the kernel of the natural map from the $\Sigma$-arithmetic completion of $B^{(1)}$ to its $\Sigma$-congruence completion; this is the $\Sigma$-congruence kernel. Centrality conjecture. The group $C(\Sigma)$ is central. The source presents this as a conjectural sufficient condition for the congruence subgroup property of the relevant $q$-arithmetic subgroup, and states that the underlying congruence subgroup problem remains open.