Poisson order-$k$ median offset bound

Let $k>1$, $\kappa=k(k+1)/2$, and let $\alpha_{k,n}$ denote the mean parameter associated with the median of the Poisson distribution of order $k$.

Median offset bound conjecture. For fixed $k>1$ and $n\ge\kappa$,

$$\alpha_{k,n}-n\in(0,1).$$

This assertion is not always true when $n<\kappa$.

For $k=1$, the corresponding bound is known; the stated claim concerns the higher-order Poisson distributions and remains open.