Higher-order Poisson–Charlier divergence lower-bound conjecture

Let $X$ be a random variable with values in $\mathbb{N}_{0}$, let $\lambda$ be the Poisson parameter, and let $C_{k}^{\lambda}$ denote the Poisson–Charlier polynomial of order $k\in\mathbb{N}$. Write $D(X\Vert\operatorname{Po}(\lambda))$ for the divergence from the Poisson distribution with parameter $\lambda$. Higher-order Poisson–Charlier lower-bound conjecture. For every such $X$ and every $k\in\mathbb{N}$,

$$D\left(X\Vert\operatorname{Po}(\lambda)\right)\geq\frac{\mathrm{E}\left[C_{k}^{\lambda}(X)\right]^{2}}{2}$$

whenever $\mathrm{E}\left[C_{k}^{\lambda}(X)\right]\leq0$. This would extend the established first-order Poisson bound to every order of the Poisson–Charlier polynomial; the source provides no resolution of the conjecture.