Hereditarily just-infinite pro-$2$ groups

Question

An infinite profinite group $G$ is called just-infinite if all non-trivial closed normal subgroups of $G$ have finite index. A profinite group is called hereditarily just-infinite if every open subgroup is just-infinite. If $p$ is an odd prime, then it's known that the pro-$p$ group ${\rm SL}_{n}^{1}(\mathbb{Z}_{p}):=\ker({\rm SL}_{n}(\mathbb{Z}_{p})\to {\rm SL}_{n}(\mathbb{F}_{p}))$ is hereditarily just-infinite where $\mathbb{Z}_p$ is the ring of $p$-adic integers. However, it is obvious that ${\rm SL}_{2}^{1}(\mathbb{Z}_{2}):=\ker({\rm SL}_{2}(\mathbb{Z}_{2})\to {\rm SL}_{2}(\mathbb{F}_{2}))$ is not just-infinite since $\{\pm I_2\}$ is a closed normal subgroup of ${\rm SL}_{2}^{1}(\mathbb{Z}_{2})$. This leads to the following questions:

1. Is ${\rm SL}_{2}^{1}(\mathbb{Z}_{2})/\{\pm I_2\}$ hereditarily just-infinite?
2. Let ${\rm SL}_{2}^{2}(\mathbb{Z}_{2}):=\ker({\rm SL}_{2}(\mathbb{Z}_{2})\twoheadrightarrow {\rm SL}_{2}(\mathbb{Z}_{2}/4\mathbb{Z}_{2})$. Is ${\rm SL}_{2}^{2}(\mathbb{Z}_{2})$ hereditarily just-infinite?
3. Is the projective special linear group ${\rm PSL}_{2}(\mathbb{Z}_{2})$ over $\mathbb{Z}_2$ hereditarily just-infinite?

Any references will be appreciated.