Elementary type conjecture for finitely generated Demushkin pro-p groups

Let $K$ be a field such that its maximal pro-$p$ Galois quotient $G_K(p)$ is a finitely generated pro-$p$ Demushkin group of rank at least $3$. Elementary type conjecture for Demushkin groups. There is a finite extension $F/\mathbb{Q}_p$ containing $\zeta_p$ such that

$$G_K(p) \simeq G_F(p).$$

This is presented as an expected consequence in accordance with the Elementary Type Conjecture. The source gives no evidence of resolution, so the assertion remains open.