2 problems

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.

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.