Blackheads can sometimes get inflamed which is part of the process of a clogged pore becoming infected with acne bacteria. That said, lots of people with blackheads don't progress to inflamed lesions. If you will or won't depends on many factors including skin type, hormones, sensitivity to acne bacteria, degree of general body inflammation, etc.
Usually inflammatory acne develops after a pore is clogged with a tiny plug that is microscopic within the pore. This tiny plug deprives the pore of oxygen which creates the perfect environment for acne bacteria to multiply. Then inflammation and infection can begin which results in visible lesions. Bigger plugs like blackheads can do it too.
This brings us to diet and Dan's regime. Diet can help the body deal with inflammation better, lower inflammation, and help to balance hormones and hormonal reactions. Consequently there is less oil so there are fewer plugs. If a pore does get clogged and infected with acne bacteria the skin's reaction is less inflammatory because the whole body is calmer and not so reactive. For some people diet alone is sufficient to control acne...skin plugs are then naturally expelled before serious inflammation sets in...hence less or no acne.
For other people diet is not enough. I am one of those people. I need some meds along with diet. However meds alone did not get me clear...adding a whole foods diet and avoiding dairy-sugar-processed fats (very similar to Piagem) finally got me clear. I went to many derms over the years (am 63 now) who said diet did not matter. But I am living proof that diet does matter, that meds are not enough for some people and a good anti-acne diet can help tip the scale to clear skin.
Dan writes about his regime helping. It does this by using BP which opens up the pores, even clogged pores, to allow oxygen in which kills acne bacteria. BP also helps dissolve the plugs. I personally can't use BP topically because I have extremely sensitive skin but a BP cleanser works great and is gentle enough for my skin. I also recommend microderms to help keep pores clear of dead skin which is part of what plugs are made of...that and oils.
P.S. Piagems --- I can not eat nuts of any kind. They clog me up every time. However other foods with natural fats are just fine for my skin (avocado, fish, etc.). It was extremely hard for me to give up nuts so I did specific experiments where I'd eat nuts and record what happened versus when I avoided nuts. For me, there was a definite increase in facial oil when I ate nuts and inflamed acne. You may want to do a similar experiment and see what happens. I hope that nuts (cashews) are not an issue for you. I know they are a big component of many vegan type diets...but if they are an issue for you there are ways to still get needed calories and make delicious meals without them.