Lets do a check. Assume a worst case scenario where nobody publishes false results at all.

To get three p < 0.05 studies if the hypothesis is false requires on average 60 experiments. This is a lot but is within the realms of possibility if the issue is one which many people are interested in, so there is still grounds for scepticism of this result.

To get one p < 0.001 study if the hypothesis is false requires on average 1000 experiments. This is pretty implausible, so I would be much happier to treat this result as an indisputable fact, even in a field with many vested interests (assuming everything else about the experiment is sound).