Deninger's exchanged-place Leopoldt conjecture

Assume that $k$ is a CM field. Let $E_p(k)$ be the group of $p$-Weil numbers in $k$, let

$$V_k=\bigoplus_{v\mid\infty}{\mathbb R}(1)_v,$$

where ${\mathbb R}(1)_v$ is the purely imaginary subspace at the complex place $v$, and let

$$\alpha_{p,\infty}:E_p(k)\longrightarrow\bigoplus_{v\mid\infty}{\mathbb R}(1)_v/{\mathbb Z}(1)_v$$

be the multivalued complex-logarithm map. If $\xi_1,\ldots,\xi_m$ is a basis of $E_p(k)\otimes {\mathbb Q}$, choose for each $i$ a lift $\eta_i\in V_k$ of $(\alpha_{p,\infty}\otimes 1)(\xi_i)$. Deninger's exchanged-place Leopoldt conjecture. For every such choice of lifts, the vectors $\eta_1,\ldots,\eta_m$ are ${\mathbb R}$-linearly independent in $V_k$. This is the strongest formulation proposed for exchanging $p$ and $\infty$ in Leopoldt's conjecture; the paper gives an equivalent formulation in terms of arguments of conjugates of Weil numbers, but no general proof.