The classification conjecture for strongly hereditarily self-similar pro-p groups

Let $p$ be a prime, and let $G$ be a torsion-free $p$-adic analytic pro-$p$ group of dimension $d$. Suppose that $p>d$. A group is strongly hereditarily self-similar of index $p$ when every open subgroup has a self-similar action of index $p$ in the strong sense used in the paper.

Classification conjecture. $G$ is strongly hereditarily self-similar of index $p$ if and only if $G$ is isomorphic to

$$\mathbb{Z}_p^d$$

for $d\geqslant 1$, or to $G^d(s)$ for $d\geqslant 2$ and some integer $s\geqslant 1$.

The statement extends the three-dimensional classification beyond the solvable case and to higher dimensions. The supplied text gives no resolution, so its status remains open.