Monotonicity of the Poisson order-$k$ median offset

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 monotonicity conjecture. The function $\alpha_{k,n}-n$ is decreasing in $n$ under one of the following conditions: (i) $k=2$ and $n\ge3\kappa$; (ii) $k\in[3,6]$ and $n\ge2\kappa$; or (iii) $k\ge7$ and $n\ge\kappa$.

The source reports counterexamples below these ranges, while the analogous assertion for $k=1$ is known to have been proved. The stated ranges remain conjectural for $k>1$.