  Recent years have witnessed the success of adaptive (or unified) approaches
in estimating symmetric properties of discrete distributions, where one first
obtains a distribution estimator independent of the target property, and then
plugs the estimator into the target property as the final estimator. Several
such approaches have been proposed and proved to be adaptively optimal, i.e.
they achieve the optimal sample complexity for a large class of properties
within a low accuracy, especially for a large estimation error $\varepsilon\gg
n^{-1/3}$ where $n$ is the sample size.
  In this paper, we characterize the high accuracy limitation, or the penalty
for adaptation, for all such approaches. Specifically, we show that under a
mild assumption that the distribution estimator is close to the true sorted
distribution in expectation, any adaptive approach cannot achieve the optimal
sample complexity for every $1$-Lipschitz property within accuracy $\varepsilon
\ll n^{-1/3}$. In particular, this result disproves a conjecture in [Acharya et
al. 2017] that the profile maximum likelihood (PML) plug-in approach is optimal
in property estimation for all ranges of $\varepsilon$, and confirms a
conjecture in [Han and Shiragur, 2021] that their competitive analysis of the
PML is tight.
