Let $K$ be a number field and $O_K$ its ring of integers. Given any non-unit element $\alpha\in O_K$, does there exist a unit $u\in O^\times_K$ such that $$ |\sigma(u \alpha)| >1 \quad \text{ for all } \sigma \in \mathrm{Hom}(K, \mathbb{C}). $$

I do not find a counter-example; and I can not see how to use Minkowski theory to find such an element in the ideal lattice $\alpha O_K \subset \prod_{v\mid \infty} K_v$.