Way to solve 5-dimentional Linear assignment problem
If you had as much people as number of people in your clusters (each people must have a dot in each cluster) then maybe you could hope for some divide-and-conquer strategy with polynomial runtime, because of greedy-like regularities. For example, in 1D and with 3 people only, after sorting the points along the axis a greedy sweeping algorithm yields an optimal solution ; with some tweaks maybe this kind of ideas could work in higher dimensions. But with some people who can not be part of a specific cluster, this doesn't seem to work anymore.
If I were you, I'd look for approximation algorithms (good ones are known for the 3d matching). Maybe some of them can yield good approximation factors with your problem's structure.