Let $\Delta$ be the set of all probability distributions over $\mathbb{N}=\{1,2,\ldots\}$ and fix some $\mathcal{P}\subseteq\Delta$.
Suppose additionally that $\Delta$ is endowed with some norm $||\cdot||$.
An *estimator* $\hat p_n$ for $\mathcal{P}$ is a mapping $\mathbb{N}^n\to\Delta$.
We say that $p_n^*$ is a *minimax optimal* estimator if
$$
\inf_{\hat p^n}\sup_{P\in\mathcal{P}} \mathbb{E}||\hat p_n(X_1,\ldots,X_n)-P||
=
\sup_{P\in\mathcal{P}} \mathbb{E}||p_n^*(X_1,\ldots,X_n)-P||,
$$
where the infimum is over all estimators
and the expectation is over $n$
independent copies of $X\sim P$.

I conjecture that $p_n^*$ can always be chosen to be of the following form: its $i$th coordinate is given by some function $f_i:[0,1]\to[0,1]$ of the empirical frequency of the $i$th symbol, $$ f_i: n^{-1}\sum_{j=1}^n1[X_j=i] \mapsto [0,1], $$ where the $f_i, i\in\mathbb{N}$ are fixed, deterministic, and entirely determined by $\mathcal{P}$ and are further monotonically nondecreasing. (For instance, if $\mathcal{P}=\Delta$ and $||\cdot||$ is either $\ell_1$ or $\ell_\infty$, then minimax optimality is attained by the Maximum Likelihood Estimator, where the $f_i$ are the identity function.)

Question: Is this true? Known?

Update: Václav Voráček provided what appears to be a counterexample: $P(1)>0\implies P(2)=c$ for some constant $c$, for all $P\in\mathcal{P}$, and $P(1)=0\implies P(2)=0$.

Hence I **additionally** assume that
$\mathcal{P}$ is permutation-invariant, in which case I conjecture that all of the $f_i$ can be taken to be identical.

Update II. I don't mind if $f$ is also a function of $n$ (in addition to the first argument, which is the empirical frequency) -- as long as it is monotone in the first argument.

Update III. Following some discussion (mostly with Václav), I'm making the conjecture more modest yet. Let's restrict $\mathcal{P}$ to those distributions $P$ for which $\sup_{i\in\mathbb{N}}P(i)(1-P(i))\le v$ and the norm $||\cdot||$ to be $\ell_\infty$.