The improved sample-independent MGF bound for binomial KL divergence

For a binomial experiment with sample size $n$, success probability $p$, empirical proportion $B/n$, and moment-generating-function parameter $x\in[0,1)$, define the binary Kullback–Leibler divergence by

$$D\bigl((u,1-u)\mathbin\Vert(v,1-v)\bigr)=u\log\frac{u}{v}+(1-u)\log\frac{1-u}{1-v}.$$

The improved binomial MGF conjecture. The function

$$f(x)=\frac{2}{\sqrt{1-x}}-1$$

is a sample-independent upper bound for the MGF of the binomial KL divergence; equivalently, for every positive integer $n$, $p\in[0,1]$, and $x\in[0,1)$,

$$\operatorname*{E}\left[\exp\left(xnD\left(\left(\frac{B}{n},1-\frac{B}{n}\right)\mathbin\Vert(p,1-p)\right)\right)\right]\leq \frac{2}{\sqrt{1-x}}-1,$$

where $B\sim\operatorname{Binomial}(n,p)$. This would improve the existing bound $1/(1-x)$ and remain compatible with the asymptotic bound $1/\sqrt{1-x}$, which is not valid uniformly for all finite samples.