Maximal number of subsets of $n$ real numbers that have the same sum, when $2^{n-2}$ of subsets have "unique" sums

Question

I've recently found a problem that I still can't solve: Dependency of the properties of numbers' subsets concerning subsets' sums of $n$ real numbers. A problem linked with it, that may be interesting not only for me has been proposed by @RossMillikan.

Having $2^{n-2}$ subsets of "unique" sums gives a hypothetical possibility that the remaining $3 \cdot 2^{n-2}$ subsets may have the same sum, which violates the given inequality (the second number is too big). What (and why) is instead the maximal number of the remaining subsets then?

I can't figure out why there may not be such a combination. I know that even for small $n=2$ sums of two subsets influence the third one, but I don't know what are the general dependencies. I would appreciate any help in this and/or linked problem.