The higher-rank generalization of Lehmer's conjecture for units

Let $F$ be a number field, let $L_1$ be the logarithmic image of $\mathcal{O}_F^*$, and for each $k$ define $\mu_{1,k}(F)$ as the minimum of the $L^1$-norms of wedges $l_1\wedge\cdots\wedge l_k$ of $k$ linearly independent elements of $L_1$. Higher-rank Lehmer conjecture. For each $k\in\mathbb{N}$ there exists an absolute constant $\delta_k>0$ such that

$$\mu_{1,k}(F)\geq\delta_k$$

for all number fields $F$ with $r\geq k$. This extends Lehmer's conjecture from individual logarithmic units to exterior powers of the unit lattice; the statement is presented as a general conjectural version and remains open in general.