HLS. 7 out of 7 of my exams 1L were takehomes exams taken either in Exam 4's Takehome or Open+Network mode, so I can only speak to that with any personal experience. Overview:
Takehome: Can access hard-drive, internet, and copy+paste.
Open+Network: Can access hard drive, internet, but can't copy+paste
Open + No Network: Can access hard drive, no internet, no copy+paste.
For takehome. I type my entire exam in word, with the doc saved to dropbox in case something catastrophic happens to my computer, and then c/p it over to Exam 4 in the last 5 minutes to submit. It's sort of just an annoying extra layer to being able to just upload a word doc.
None of the profs I had vaguely considered Closed since they figured it would just provide an advantage to the people who cheated (since it was takehome, you could easily just have another laptop with you). For one of my classes, students anticipated the exam question (correctly), pre-wrote their answer over several days, copy-pasted their answers in, and called it a day.