The subexponential subgroup growth conjecture for pro-$p$ groups

Let $G$ be a finitely generated pro-$p$ group. It has subexponential subgroup growth if, writing $a_n(G)$ for the number of open subgroups of index $n$, one has

$$\limsup_{n\to\infty}a_n(G)^{1/n}=1.$$

Let $\overline{\chi}_2(G,\mathbb{F}_p)$ denote the relevant normalized second Euler characteristic of $G$.

Subexponential subgroup growth conjecture. If $G$ has subexponential subgroup growth and

$$\overline{\chi}_2(G,\mathbb{F}_p)=0,$$

then $G$ is isomorphic to $\mathbb{Z}_p$ or to $\mathbb{Z}_p\rtimes\mathbb{Z}_p$.

This conjecture proposes a classification of finitely generated pro-$p$ groups combining restricted subgroup growth with vanishing normalized second Euler characteristic. The supplied text presents it as a belief, and gives no resolution.