I found out today the standard terminology for my favorite conjecture -
This directly implies that the 4SUM problem is quadratic. I believe that the 3SUM problem is also quadratic (as does everybody), but that the reasons for that are much deeper and more complex.
It also implies P != NP, so we can rest assured that noone has proven it yet.
Given the obviousness of this conjecture I'm sure circuit complexity people have already spent considerable time on it and just haven't gotten anywhere, although curiously googling for '3sum' and 'circuit complexity' doesn't turn up anyone else musing on the subject.
Here is a good list of open problems to spend time on.