A finiteness theorem for a class of exponential congruences

For given elements $\alpha _1,\ldots,\alpha _k$ and $\beta$ belonging to the ring of integers $\mathcal{A}$ of a number field we consider the set of all $k-$tuples $(a_1,\ldots,a_k)$ in $\mathbb{N}^k$ for which $\sum _{i=1}^{k}\alpha _i\beta^{a_i}$ divides $\sum _{i=1}^{k}\alpha _i z^{a_i}$ for any $z\in\mathcal{A},$ and prove under some reasonable assumptions that the set of solutions is finite.