Distance magic labeling in complete 4-partite graphs

Let $G$ be a complete $k$-partite simple undirected graph with parts of sizes $p_1\le p_2\mathrm{.}\mathrm{.}\mathrm{.}\le p_k$. Let $P_j={\sum }_{i=1}^j{p}_i$ for $j=1,\mathrm{.}\mathrm{.}\mathrm{.},k$. It is conjectured that $G$ has distance magic labeling if and only if ${\sum }_{i=1}^{P_j}(n-i+1)\ge j\left(\genfrac{}{}{0px}{}{n+1}{2}\right)\mathrm{/}k$ for all $j=1,\mathrm{.}\mathrm{.}\mathrm{.},k$. The conjecture is proved for $k=4$, extending earlier results for $k=2,3$.