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Or overestimate?

Holding all else equal, noisier estimates bias us towards the null. This is attenuation bias.

However, the estimates are still probably overestimated. Confounding, p-hacking, publication bias, all move us towards larger estimates.

“Still probably” is classic statistical science

I would think that, by default, noise would not have a bias? Adding noise doesn't change the mean, it just increases the variance, right?

It pushes confidence bounds closer to the null hypothesis.