The partial local p-th power conjecture for Galois extensions

Let $K/\mathbb{Q}$ be a Galois extension of degree $n$ with Galois group $G$, and let $\eta\in K^{\times}$ be such that the $\mathbb{Z}[G]$-module generated by $\eta$ has $\mathbb{Z}$-rank $n$. The partial local p-th power conjecture. For every sufficiently large prime $p$, $\eta$ is not a partial local $p$-th power at $p$. The conjecture is designed to exclude the nontrivial vanishing of local $\theta$-regulators; the source states that no particular case of these stronger conjectures appears to be known.