  For a continuous map $T: X\rightarrow X$ on a compact metric space $(X,d)$,
we say that a function $f: X \rightarrow \mathbb{R}$ has the property
$\mathscr{P}_T$ if its time averages along forward orbits of $T$ are maximized
at a periodic orbit. In this paper, we prove that for the one-side full shift
of two symbols, the property $\mathscr{P}_T$ is prevalent (in the sense of
Hunt--Sauer--Yorke) in spaces of Lipschitz functions with respect to metrics
with mildly fast decaying rate on the diameters of cylinder sets. This result
is a strengthening of \cite[Theorem~A]{BZ16}, confirms the prediction mentioned
in the ICM proceeding contribution of J. Bochi (\cite[Seciton 1]{Boc18})
suggested by experimental evidence, and is another step towards the Hunt--Ott
conjectures in the area of ergodic optimization.
