Just to add to the confusion, it is well documented that obesity is associated with metabolic inflexibility (the inability to change fuel source) which is possibly why the first few fasts are so hard for some people. Further, a study on Ramadan fasting showed that some people entered ketosis fairly soon after the start of Ramadan but others (who were metabolically inflexible) never appeared to enter ketosis, suggesting that for them, the repeated fasting did not increase their ability to burn fat (or at least their ability to manufacture ketones anyway).

It is this great individual variability that causes some people to get big changes in cholesterol (and blood pressure and other health indicators) with weight loss/fasting/low carb but others don't.