Most (if not all) nuclear scientists and radiation experts I've seen tslking about this say this is a wrong assumption. Cheryl Rofer, former nuclear scientist, made a napkin calculation and suggested it would need about 57 years of camping in the hottest (= most irradiated) parts of the red forest to acquire ARS (acute radiation syndrome). Even if the digging and kicking up dust there elevated the radioactivity levels, it would take a hell of a lot to get from 57 years to one month. Keep in mind that the most radioactive snd thus dangerous elements also decay the quickest, because of course both properties stem from the same thing, redioactive decay. The more dangerous the quicker it's gone.
Radiation in the red forest is supposedly not terrible _ if you don't disturb the soil _
All the nastier bits are under a layer of "clean" soil and you would have no contact with them ordinarily.
These soldiers actually dug down to the nasty bits and likely breathed some of the dust in.
Between that, and eating wildlife (which acts as a powerful concentrator), I'm not surprised they died. The exposure must have been thousands of times above the already non-insignificant baseline.