Infinitesimal containment conjectures for congruence completions

In the first case, let $\Gamma=\operatorname{SL}_2(\mathbb{Z})$ and let $p$ be any prime. In the second case, let $\Gamma$ be a cocompact arithmetic lattice of the second type in $\operatorname{SL}_2(\mathbb{C})$, so that $\Gamma$ does not intersect any conjugate $H$ of $\operatorname{SL}_2(\mathbb{R})$ as a lattice in $H$, and let $\widehat{\Gamma}^{c}$ be its congruence completion. Congruence-completion infinitesimal containment conjectures. In the first case,

$$(\operatorname{SL}_2(\mathbb{Z}_p),\mu)\xhookrightarrow{\operatorname{inf}}\operatorname{SL}_2(\mathbb{Z}).$$

In the second case,

$$(\widehat{\Gamma}^{c},\mu)\xhookrightarrow{\operatorname{inf}}\Gamma.$$

These are further specializations of the non-amenable image criterion for compact actions, and the source gives no resolution.