Efrat's arithmetical Elementary Type Conjecture

Let $F$ be a field containing a root of unity of order $p$ and such that $G_F(p)$ is finitely generated. A free pro-$p$ product decomposition is an expression

$$G_F(p)=G_1*_p\cdots*_pG_n.$$

Efrat's arithmetical Elementary Type Conjecture. There is such a decomposition where, for each $1\leq i\leq n$, one of the following holds: $G_i\cong\mathbb{Z}_p$; $G_i$ is the decomposition group of some extension $\tilde v$ of a valuation $v$ on $F$ to $F(p)$ with nontrivial inertia group; or $p=2$ and $G_i\cong\mathbb{Z}/2$. This conjecture is the arithmetical form of the Elementary Type Conjecture, describing finitely generated maximal pro-$p$ Galois groups through free products of elementary building blocks and valuation-theoretic decomposition groups. Its resolution status is not specified in the supplied text.