Centered gamma moment-generating-function 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 $P$. Let $t$ be a real parameter satisfying $0\leq t<n$.

Centered gamma moment-generating-function conjecture. For every $n$, $k$, and $P$,

$$\log\mathbb{E}\left[\exp\left(t\left(V_{n,k,P}-\mathbb{E}[V_{n,k,P}]\right)\right)\right] \leq (k-1)\log\left(\frac{\exp(-t/n)}{1-t/n}\right).$$

The right-hand side is the centered log moment-generating function of a gamma distribution with shape $k-1$ and rate $n$. The conjecture would imply a one-sided form of the preceding concentration conjecture; it was attributed in the source to Agrawal and remains unresolved here.