This was essentially answered by Nate in the comments, but here are some more details.  As Nate argues, $|m_i - m_j| \leq 1$ for all distinct $i,j$.  Thus, if $s=ak+r$, where $a,r \in \mathbb{N}$ and $r < k$, then there is a unique choice (up to permuting variables) which maximizes the product.  Namely, set $r$ of the variables to $\lceil s/k \rceil$ and the rest to $\lfloor s/k \rfloor$.  

This problem is related to [Turán's Theorem][1], which concerns the maximum possible number of edges in a graph on $s$ vertices with no $K_{k+1}$ subgraph.  The answer is given by the Turán graph, which is unique.  More generally, Zykov proved that among $K_{k+1}$-free graphs, the Turán graph also has the most number of complete graphs $K_t$ for all $t \leq k$.  The case $t=k$ leads to your optimization problem (once you have already established that such a graph must be $k$-partite).  


  [1]: https://en.wikipedia.org/wiki/Tur%C3%A1n%27s_theorem