Power paradox: overestimated effect size in low-powered study, but the estimator is unbiased If we have an underpowered study but manage to reject the null hypothesis, anyway, it makes sense to wonder if we have overestimated the effect size.
However, such a concern seems unwarranted if we use an unbiased estimator of the effect size (such as $\bar X_1-\bar X_2$ to estimate $\mu_1-\mu_2$).
At the same time, there is an appeal to speculating that we must have lucked into a particularly large observed effect to have managed to reject despite the low power.
This seems like a paradox. Does it have a resolution?

It's not a paradox. You may call it a dilemma, or more precisely an unknown. You have correctly narrowed it down to the two possible outcomes: you are either really "lucky", or the assumptions behind the power calculation are incorrect. There is no way to know which is which based on the results of one study alone. These considerations matter even for well powered studies with statistically significant findings.
