Cancer in Sweden from Chernobyl, or, Why the LNT Needs a Swift Kick

A completely bogus study from Sweden purporting to demonstrate that people are more likely to get cancer if exposed to pre-industrial levels of radiation than if they were exposed to atomic bomb detonations once again raises a question about anti-nuclear groups' unwavering support of the politically-motivated Linear-No-Threshold radiation impact hypothesis (LNT):

Why on Earth do they support it?

The LNT is, at its core, an assumption. A graph is constructed, with radiation exposure on the horizontal axis and cancer deaths on the vertical axis. The Japanese atomic bombing survivors' exposure and cancer data is then plotted, and a line is drawn from there to zero radiation and zero cancer.
Seriously.
No low-dose data is included--not from pre-industrial cancer rates, which involved basically the same radiation exposure as today yet were practically nonexistent, and not from the definitive study on the matter, which tracked all the radiation ever received by maintenance crews in Navy nuclear shipyards--and found a mortality rate over 20% lower than their coworkers in non-nuclear shipyards.

But if anti-nuclear groups want a scientific investigation of the health effects of radiation, as they so often claim to do, why do they support an assumption?

Pro-nuclear, anti-nuclear, and in-between organizations all should recognize the fallacy of relying on assumptions, and insist on a well-funded NAS investigation of the health effects of ionizing radiation, with the objective of identifying whether or not there is a threshold, and if there is one, upper and lower bounds.
I call on anti-nuclear groups to show that they have confidence in their claims by submitting them for rigorous peer review.