The liver oil, including omega-3, is the new gold, said the campaigners. Deep-sea sharks are being targeted as they make up for a lack of a swim bladder by accumulating oil in their livers to regulate their buoyancy.
The liver of the goblin shark, living at depths of up to 4,000ft and known for its bizarrely shaped head, makes up a quarter of its body weight.
Fears for these fish species come amid concern that some of the benefits of omega-3 have been oversold. The fats are credited with staving off ills from heart attacks to dementia and depression.
Studies, however, have questioned their effect on the heart. The most recent, in the Journal of the American Medical Association, examined 20 studies involving almost 70,000 people.
The surveys Greek authors found that those who took omega-3 were no less likely to die of a heart attack or stroke than others. A second-round up, by South Korean researchers in May, reached a similar conclusion.
Two years ago, the New England Journal of Medicine reported that heart attack patients taking omega-3 were no less likely to have a second attack than those who did not.
Dr Carrie Ruxton, who speaks for the supplements sector, said omega-3 can help the heart but has to be part of a healthy lifestyle.