Rodriguez Villegas's exterior-product conjecture for logarithmic embeddings

Let $k$ be a number field, let $S_{\infty}$ be the set of its archimedean places with $|S_{\infty}|\geq 2$, and let $\Lambda_{S_{\infty}}(k)$ be the logarithmic embedding, of rank

$$r=\operatorname{rank}\Lambda_{S_{\infty}}(k).$$

For vectors in this space, write $\|\boldsymbol\alpha_1\wedge\cdots\wedge\boldsymbol\alpha_q\|_1$ for the $1$-norm of their exterior product. Rodriguez Villegas's conjecture. There exist absolute constants $c_0>0$ and $c_1>1$ such that, whenever $1\leq q\leq r$ and $\boldsymbol\alpha_1,\dots,\boldsymbol\alpha_q$ are linearly independent points of $\Lambda_{S_{\infty}}(k)$,

$$c_0c_1^q\leq\|\boldsymbol\alpha_1\wedge\boldsymbol\alpha_2\wedge\cdots\wedge\boldsymbol\alpha_q\|_1.$$

For $q=1$ this contains Lehmer's problem, while the case $q=r$ is the known lower bound for the regulator proved by Zimmert; the conjecture interpolates between these two results.