Similarly, here is the situation about being overweight: AFTER menopause (and I honestly don't know about chemically-induced menopause), fat cells manufacture small levels of estrogen. To a significantly lesser extent pre-menopause, so do all fat cells. However, this is "overweight" not "healthy weight" -- so I don't know the BMI or anything.
Here it is, courtesy of Mayo Clinic:
25.0 — 29.9 Overweight
30.0 and higher Obese
The risk is overweight as measured by BMI, not by high fat diet:
Research shows that eating a diet rich in fruits and vegetables doesn't offer direct protection from breast cancer. In addition, a recent study of dietary fat and breast cancer showed only a slight decrease in the risk of invasive breast cancer for women who ate a low-fat diet.