Concentration conjecture for empirical relative entropy

Let $V_{n,k,P}$ denote the empirical relative entropy statistic for a multinomial distribution with sample size $n$, alphabet size $k$, and distribution $P$. For $\theta\geq 0$, write

$$\theta=\varepsilon.$$

Concentration conjecture. There are positive constants $c_1$ and $c_2$ such that for every $n$, $k$, $P$, and $\varepsilon\geq 0$,

$$\Pr\left\{\left|V_{n,k,P}-\mathbb{E}[V_{n,k,P}]\right|\geq\varepsilon\right\}\leq c_1\exp\left\{-c_2\min\left\{\frac{n^2\varepsilon^2}{k-1},n\varepsilon\right\}\right\}.$$

This conjecture would give finite-sample two-sided concentration of empirical relative entropy around its mean, improving bounds that focus only on deviations above zero. It was posed in the cited work and remains unresolved here.