Demuškin-to-$p$-adic type conjecture

Let $F$ be a field containing a root of unity of order $p$. A Demuškin cyclotomic pro-$p$ pair is the pair $\mathcal{G}_F(p)$ whose underlying maximal pro-$p$ Galois group is Demuškin; its rank is the dimension of its first cohomology over $\mathbb{F}_p$. The pair has $p$-adic type when it is the cyclotomic pro-$p$ pair associated with a finite extension of $\mathbb{Q}_p(\mu_p)$.

Demuškin-to-$p$-adic type conjecture. If $\mathcal{G}_F(p)$ is a Demuškin cyclotomic pro-$p$ pair of rank at least $3$, then $\mathcal{G}_F(p)$ has $p$-adic type.

Every cyclotomic pair of $p$-adic type has both Galois type and Demuškin type, but the converse implication is stated to be open for cyclotomic pairs of Galois type that are not already of $p$-adic type. The conjecture would exclude many Demuškin pro-$p$ groups from occurring as maximal pro-$p$ Galois groups.