Demuškin maximal pro-$p$ Galois group conjecture

Let $F$ be a field containing a root of unity of order $p$. The field $F$ is arithmetically Demuškin when it satisfies the arithmetical conditions defined earlier in the paper. A pro-$p$ Demuškin group is a pro-$p$ group with Demuškin structure, and $G_F(p)$ denotes the maximal pro-$p$ Galois group of $F$.

Demuškin maximal pro-$p$ Galois group conjecture. If $G_F(p)$ is a pro-$p$ Demuškin group of rank at least $3$, then $F$ is arithmetically Demuškin.

This is presented as the converse of the stated theorem that an arithmetically Demuškin field has a maximal pro-$p$ Galois group of $p$-adic type. The conjecture seeks an arithmetic characterization of those fields whose maximal pro-$p$ Galois groups are Demuškin.