Let $P=(p_1,\ldots,p_d)$ be a distribution on $[d]$. Given $n$ iid draws from $P$, we construct some empirical estimate $\hat P_n=(\hat p_{n,1},\ldots,\hat p_{n,d})$. Let us define the $r$-risk by $$ J_n^r = \sum_{i=1}^d |p_i-\hat p_{n,i}|^r. $$
It is known (see, e.g., Lemma 2.4 here) that when $\hat P_n$ is the maximum likelihood (i.e., empirical frequency) estimator and $r\ge2$, we have $\mathbb{E}[J_n^r]\le1/n$. In particular, the expected $r$-risk decays at a dimension-free rate.
It is also known that for $r=1$, the risk decays at a minimax rate of $\Theta(\sqrt{d/n})$.
Question: what is known for $1<r<2$?