Serre's congruence subgroup property conjecture

Let $K$ be a global field, let $S$ be a finite set of places, and retain the paper's notation for the corresponding arithmetic group $({\mathbf{G}},S)$. The pair has the congruence subgroup property (CSP) when its congruence kernel is finite. Serre's conjecture. The pair $({\mathbf{G}},S)$ has CSP if ${\mathbf{G}}$ is $K_\rho$-isotropic for every $\rho\in S-\mathcal{V}_\infty$ and

$$\sum_{\rho\in S}\operatorname{rank}_{K_\rho}G>1.$$

The pair $({\mathbf{G}},S)$ does not have CSP if

$$\sum_{\rho\in S}\operatorname{rank}_{K_\rho}G=1.$$

This is identified in the source as the main conjecture in the congruence subgroup problem. The supplied text gives no resolution status, so it is recorded as open.