the non-scientific answer from me is that it's ridiculous. water fast, apple fast, whatever other stupid things are out there... idk who comes up with them. your body is an engine and it needs energy. i think you're looking for a quick fix or shortcut, and there just aren't any. you have to examine your diet, figure out your triggers, and avoid them. i'm sure alternativista will have a legit answer for you.
you may improve over those three days because you are inadvertently avoiding your dietary triggers, but then when you binge on the fourth day it'll be back to business as usual - you won't be healed. i do IF because i am cutting weight and it makes it easier for me. i can't comment on it's effects on acne for me, although i've read that it has benefits.
bunch of gluten free rice chex + whole milk (only time i drink milk currently)
3 protein shakes a day (1 scoop each with water)
2 meals of 1/2-3/4 lb beef or chicken + rice (now brown) or gluten free pasta + sauce/etc. to get to >1k cals per meal
big sweet potato (just microwaved with a little cinnamon and brown sugar butter + cinnamon)
macadamia nuts ('low' in omega 6, used to eat tons of peanuts which are way high in o6) during the day between classes
I was on antibiotics (minocycline and tetracycline) for about a year i think. currently on my second bottle of probiotics which have really helped get my digestion back to normal, but idk if my acne was affected.
This x10. some people find the most random things that cause their acne, just have to keep experimenting.
bleh i was typing up a big thing but it ended up getting way off topic lol.
something in your 'healthy' diet still isn't working for you. maybe post exactly what you eat day to day?
for instance, i've been eating as healthy as i can (balanced omega 3/6, gluten free, very little sugar, no fastfood/takeout/snack/processed food) while still getting 3.5-4k cals a day and my acne has significantly improved but still not perfect yet. at the end of last week i switched white rice for brown and i'm significantly clearer now. so that little change to my already 'healthy' diet might be the answer for me (hopefully!!).
i've never waited that long, even for huge ones (just had one this week). what i do is ice it as soon as it pops up, then a few layers of tea tree oil (i do a 1:5 dilution with water, might step up to 1:3), then just cake on the BP (10%). I had one pop up on me late sunday night (the size of a quarter, already too big for ice, don't know how i missed it...). I did what I just told you and overnight it got really hard and sorta came to a head. i was able to pop it and drain most of the oil the following night. then just more tea tree oil and BP day and night until it dries up. It's a full scab now and should be good and healed hopefully by friday or saturday night (putting neosporin on now). I went pretty crazy getting the oil out, most probably would do that (hurts), and it might leave a scar (doesn't for me). but i would definitely try the tea tree oil and BP combo, that seemed to bring it to a head overnight and the dry it up.