The correct study is one where some participants get an intervention to make them exercise more, and some are left alone. There's no ethical issue, but running

The correct study is one where some participants get an intervention to make them exercise more, and some are left alone. There's no ethical issue, but running such a study would be difficult and expensive. Adjusting for BMI does not solve the problem, because there are many things besides BMI (unmeasured confounders) which affect both cancer risk and willingness to exercise.