I did a load of research into metabolism over the last 18 months, and during my travels into understanding basic metabolism as currently demonstrated by recent scientific studies, it became quite clear to me that there are two types of fat involved in obesity. Carb metabolism ends up with both glucogen and water being stored in muscle tissue cells. Controlling carb intake directly affects these stores, but does not affect the second set of fat stores, namely the adipocytes in the brown or white fat cells.
These are controlled primarily by fat intake (and Fructose), not by carbs at all, and it is these cells that once created will always remain as a permanent fat store, We can deplete them somewhat by exercise or by following a LC diet that causes ketogenesis to occur, thus burning up the fat. BUT a low fat diet is not very successful at reducing this fat layer as countless dieters can avow to. So calorie restriction diets can reduce the glucogen stores and heavy exercise can reduce adipose stores, but its not very successful, and putting weight back on is sadly too easy,
I use an LCHF diet myself, and I have lost about 5" off my tum, and gone from 84kg down to 65kg. I have lost more than that and have had to put weight back on. I do that by increasing my fat intake, and this last 3 months I have kept to 66kg on successive weighings. So yes, I do believe fat intake does significantly affect my BMI, and I have demonstrated this and posted graphs of my results elsewhere on the forum. I do not think I am unique in this