Let $K$ be a number field with $[K : \mathbf{Q}] = d$, and let $p$ be a prime. Let $\sigma_1, \dots, \sigma_d$ be the embeddings of $K$ into $\mathbf{C}_p$. Let $u_1, \dots, u_k$ be a basis of $\mathcal{O}_K^\times$ modulo torsion, so $k = r_1 + r_2 - 1$ where $r_1$ and $r_2$ are the number of real and complex infinite places.

Consider the $d \times k$ matrix $M$ whose $i, j$ entry is $\log \sigma_i(u_j)$. Leopoldt's conjecture claims that $M$ has full rank, i.e. it has some $k \times k$ minor which is non-singular.

I'm wondering about the following conjecture:

If $K$ does not contain a CM field, then all $k \times k$ minors of $M$ are nonsingular.

Obviously this is stronger than Leopoldt's conjecture, so there is no hope of proving it; but does anyone know if such questions have been studied? Are counterexamples known?

(The statement is definitely false if $K$ contains a CM field $E$ with $[E : \mathbf{Q}] > 4$.)