Dubickas conjecture on multiplicative dependence of shifted algebraic numbers

For an algebraic number $\alpha$, consider the set ${\mathbb N}+\alpha=\{n+\alpha:n\in{\mathbb N}\}$. A set of nonzero algebraic numbers is multiplicatively independent if no nontrivial product of its elements raised to integer powers equals $1$. Dubickas conjecture. There exists some algebraic number $\alpha$ such that ${\mathbb N}+\alpha$ is multiplicatively independent. This conjecture is presented as the opposite answer to the question of whether ${\mathbb N}+\alpha$ is multiplicatively dependent for every algebraic number $\alpha$. The question has an affirmative answer for quadratic number fields, and for cubic number fields when shifts by rationals are allowed, but its general status is not given here.