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linear no-threshold is weird & unusual for most processes, but it meshes perfectly well with what I understand about radioactivity in general- that is, random & cumulative. See half lives for example.

Is there any real evidence showing LNT is wrong?

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Yes. The human body has DNA repair mechanisms that work fine if not overwhelmed. There's also lots of data showing low cancer rates despite higher-than-normal radiation exposure; e.g. some cities have quite high naturally occurring radiation, but low cancer rates.

Ok, so LNT may describe the damage but not necessarily the all-cause mortality. I can live with that.

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