Bertrand–Rodriguez Villegas conjecture for exterior powers of units

Let $L$ be a number field, let $cmathcal{A}_L$ be its set of Archimedean places, and define

$$\operatorname{LOG}:{\mathcal{O}_L^*}\longrightarrow\mathbb{R}^{\mathcal{A}_L},\qquad (\operatorname{LOG}(\gamma))_v=e_v\log|\gamma|_v,$$

where $e_v=1$ for real $v$ and $e_v=2$ for complex $v$. For each positive integer $j$, equip $\bigwedge^j\mathbb{R}^{\mathcal{A}_L}$ with the 1-norm obtained from the standard exterior basis: if $w=\sum_I c_I\delta^I$, then $\|w\|_1=\sum_I|c_I|$.

Bertrand–Rodriguez Villegas conjecture. There exist absolute constants $c_0>0$ and $c_1>1$ such that, for every number field $L$, every $j\in\mathbb{Z}_{>0}$, and every nonzero

$$w\in\bigwedge^j\operatorname{LOG}({\mathcal{O}_L^*})\subset\bigwedge^j\mathbb{R}^{\mathcal{A}_L},$$

one has

$$\|w\|_1\geq c_0c_1^j.$$

The conjecture extends Lehmer's conjecture and Zimmert's regulator bound. Results for totally real fields establish the inequality for pure wedge products of units, but the general assertion must also cover nonzero elements that are not pure wedge products; this is the important unresolved part.