Centralized concentration conjecture for empirical relative entropy

Let $P$ be a distribution on a finite alphabet of size $k$, let $\hat{P}_{n,k}$ be its empirical distribution from $n$ independent samples, and let $D(\hat{P}_{n,k} \| P)$ denote the relative entropy. There exist constants $g_1>0$ and $g_2>0$ such that, for every $t>0$, Centralized concentration conjecture.

$$\mathbb{P}\left(\left|D(\hat{P}_{n,k} \| P)-\mathbb{E}[D(\hat{P}_{n,k} \| P)]\right| \geq t \right) \leq g_1 e^{-g_2 \min\left\{ \frac{n^2t^2}{k-1}, nt \right\}}.$$

This conjecture is motivated by the asymptotic chi-squared behavior of $2nD(\hat{P}_{n,k}\|P)$ and standard concentration bounds for sub-exponential random variables. The source does not indicate that it has been proved or disproved.