The complete multipartite $\mathbb{Z}_{2^n}$-distance-magic conjecture

Let $G=K_{n_1,n_2,\ldots,n_t}$ be a complete $t$-partite graph such that

$$2^n=\sum_{i=1}^t n_i$$

and $n_i\geq 2$ for all $i$. A $\mathbb{Z}_{2^n}$-distance-magic labeling is a bijection from the vertices of $G$ to $\mathbb{Z}_{2^n}$ for which all vertex weights, computed in the group, are equal. Complete multipartite $\mathbb{Z}_{2^n}$-distance-magic conjecture. The graph $G$ is $\mathbb{Z}_{2^n}$-distance magic. The paper proves this conjecture in the stated setting.