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Unfortunately the history of science and of mathematics shows that good faith is a deceptive strategy.

I remember some years ago the announcement of a proof of the Riemann hypothesis in mathematics. Of course there have been many wrong
announcements before. But this time it was made by an eminent
mathematician, a leading figure in his field.

The press reacted enthousiastically, but the critical colleagues pointed out several gaps in the presented scetch of the proof. The colleagues were right. The Riemann hypothesis is open until today.

On the other hand it is a good strategy to be sceptical if a lay person presents a proof or a counter-example concerning a long standing conjecture. One often recognizes that the author does not make much use of the technical literature, apparently he does not know it.

Hence I recommend to look at the paper, at the name of the author and on his previous publications. If possible one should form an own opinion. But knowing the arguments of institutions like the corresponding academies and reading one or two original assessments of the experts in the field is also a good strategy.