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 seems to me that an underpowered study by definition is unlikely to give a small p-value against the null. Consequently, if you do get a small p-value it is likely that you are overestimating the true effect size.
However, if you look at all estimates from repeated experiments, regardless of their significance threshold, you do get an unbiased overall estimate. This should reconcile your paradox? Here's a simulation to illustrate:
We repeat an underpowered experiments 10,000 times:
set.seed(1234)
es <- 0.1 # The true difference
n <- 5    # and a small sample size
est <- rep(NA, 10000)
p <- rep(NA, length(est))
for(i in 1:length(est)) {
    a <- rnorm(n, mean=0)
    b <- rnorm(n, mean=0 + es)
    tt <- t.test(a, b)
    est[i] <- diff(tt$estimate)
    p[i] <- tt$p.value
}

If you consider only experiments with p < 0.05 you get extreme estimates (blue line is the true value). Note that some estimates are extreme and are also in the wrong direction (those on the left of the blue line):
hist(est[p < 0.05], xlab='Estmates where p < 0.05', main='')
abline(v=es, col='blue', lty='dashed')


Nevertheless the estimatator is unbiased across the 10,000 experiments:
mean(est)
[1] 0.1002

# Count of over- and under-estimating experiments:
length(est[est > es])
[1] 5056
length(est[est <= es])
[1] 4944

