Ozaki's non-freeness conjecture for unramified pro-$p$ Galois groups

Let $k/\mathbb{Q}$ be a finite extension and $p$ a prime number. Let $\mathcal{G}(k^\mathrm{c}_\infty)$ be the Galois group of the maximal unramified pro-$p$-extension of the cyclotomic $\mathbb{Z}_p$-extension $k^\mathrm{c}_\infty$.

Ozaki's non-freeness conjecture. The group $\mathcal{G}(k^\mathrm{c}_\infty)$ is not a non-abelian free pro-$p$-group.

The source attributes this conjecture to Ozaki and notes that Greenberg's generalized conjecture implies it when $p$ splits completely in $k$; it remains open in general.