But I fear, from what you wrote, that you fell into the same trap as @gmarx above. The claim in the exam is that the total coordination cost is above quadratic

But I fear, from what you wrote, that you fell into the same trap as @gmarx above. The claim in the exam is that the total coordination cost is above quadratic, i.e. proportional to N^b for some "b" strictly larger than 2. (Because per-capita cost is proportional to N^a for some "a" strictly larger than 1.) This is not what is typically assumed in complexity theory.