# Underpowered studies and minimum effect size

We know that that underpowered studies lead to overestimated effect sizes (conditioned on statistical significance). See here, here, and here for discussions on this topic. Now, let's suppose that the true effect size $e$ is much smaller than the minimum effect size $E$ that we use in the power calculations. As I understand it, this is not necessarily bad, if we're only interested in detecting effect of size $E$ or larger. What this means is that my study is powered appropriately to detect $E$, but underpowered to detect $e$. So my question is: is the effect estimate from this study likely to be overestimated (because we're underpowered w.r.t $e$), or is it fine (because we're well powered w.r.t $E$, which is the effect size we care about)? If the answer is the former, then why it is acceptable to specify a minimum effect size? Wouldn't we always want to specify (our best estimate of) the true effect size?