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I read somewhere that this experiment has sensitivity of 0.2 eV (I could be wrong here so please correct me).

What is this 0.45 eV then? Are they still running the measurements and trying get to 0.2 eV or this is maximum they can achieve?

[ I misread the original article explaining their equipment and project - their goal is to go all the way down to 0.2ev ]

I think it's an upper limit of around 0.2 eV^2, so √(0.2 eV^2) = 0.45 eV.

I'd like to see an experiment that's sensitive enough to eliminate the m=0 hypothesis. Their prediction of m^2 = -0.14eV^2 is a bit disconcerting.

How can we prove that one of neutrinos is massless? That would be impossible - but I do not understand these things so I can be wrong.

I believe GP is proposing the opposite -- a test capable of proving it is not massless. (Presumably this has never been done.)