Playing a little bit with Groessencharacters a stumbled on the following question:

Let $K$ be a non CM number field with $r_1$ real embeddings and $2r_2>0$ complex embeddings. Set $r=r_1+r_2-1$. Let $\epsilon_1,\ldots,\epsilon_{r}$ be a system of $\mathbf{Z}$-linearly independent (in the multiplicative notation) units of $\mathcal{O}_K$. For $x\in K$ and $j\in\{1,2,\ldots,r_2\}$ let $x^{(j)}$ be the image of $x$ under the $j$-th complex embedding (where the $r_2$ choices of complex embeddings are made so that no complex embedding are related by complex conjugation).

Q: Is it possible to choose $K$ and $\{\epsilon_1,\ldots,\epsilon_{r}\}$ so that the above assumptions are satisfied and such that there exists an $(r+r_2)$-tuple $((m_1,\ldots,m_r);(n_1,\ldots,n_{r_2}))\in\mathbf{Z}^{r+r_2}\backslash\{0\}$ such that $$ 2\pi m_j-\sum_{k=1}^{r_2} n_k\cdot\arg\left(\frac{\epsilon_j^{(k)}}{|\epsilon_j^{(k)}|}\right)=0, $$ for all $j\in\{1,\ldots,r\}$ ?

This looks like a delicate question about the $\mathbf{Q}$-linear independence of the real numbers $\arg\left(\frac{\epsilon_j^{(k)}}{|\epsilon_j^{(k)}|}\right)$.