Why are women even less represented in engineering than in other STEM?

I suspect that this is yet another example of the fractal nature of the gender binary. STEM may be coded masculine in general, but within STEM different fields are coded "more masculine" than others.

I'm very familiar with this in my own field: IT in general is masculine, but UI/frontend is more feminine than backend, on the backend system stuff is more masculine than application stuff, writing it in C is more masculine than writing it in ruby, ... This is sometimes even explicitly expressed as "your beard has to be this long to write/understand this code"

I've never found a good explanation for most of these associations, they seem fairly random to me (and apparently change between cultures as well).

There is a large body of research dedicated to underrepresentation of women in STEM.

Let me direct you to this wonderful synthesis on the topic by Wang. His work is pretty comprehensive on the subject:

Wang, M. T., & Degol, J. L. (2017). Gender gap in science, technology, engineering, and mathematics (STEM): Current knowledge, implications for practice, policy, and future directions. Educational Psychology Review, 29, 119-140.

He has also written a widely cited work here:

Wang, M. T., Eccles, J. S., & Kenny, S. (2013). Not lack of ability but more choice: individual and gender differences in STEM career choice. Psychological Science, 24, 770–775. doi: 10.1177/0956797612458937.

Rong Su produced a great meta-analysis on this topic as a graduate student:

Su, R., Rounds, J., & Armstrong, P. I. (2009). Men and things, women and people: a meta-analysis of sex differences in interests. Psychological bulletin, 135(6), 859.

If you are looking for more of a cognitive reason, you might be interested in looking at the work by Camilla Benbow, David Lubinski (both out of Vanderbilt), or Jon Wai out of Arkansas.

For example, a nice little cognitive work on gender differences recently came out in Intelligence while Jon Wai was working at Duke-

Wai, J., Hodges, J., & Makel, M. C. (2018). Sex differences in ability tilt in the right tail of cognitive abiltiies: A 35-year examination. Intelligence, 67, 76-83.