Yes. First, the 4:12 mile record is greatly inferior to the 3:50 1500m record, so you can already see that someone has run about 4:08.4 (1.08x conversion) and so it would require an 8.5 second drop to do it. That is doable. The women's 800m record is 1:53.28, and the future sub-4 woman would likely be more of a 1:54 athlete, so it would be very challenging, but we see Jakob run 3:46 (1:53 880y) off of a 1:46-47 800 pr, so a 6-7 second slowdown from 800 to 880(in a mile) is possible.
Second, the physiological advantages that men have can be gained to some degree by women in various ways, both natural and unnatural, without any new inventions or training techniques. You might have a woman on the high side of the T-limits required to do this, and of course doping was very likely not only in the women's 800m record but also the 1500m record.
Third, sub-3:50 could happen this year with Faith Kipyegon with her perfect form.