Exactness of the $k$-dependent terms in Poisson order-$k$ median asymptotics

Let $k>1$, $\kappa=k(k+1)/2$, and let $\alpha_{k,n}$ denote the mean parameter associated with the median analysis, with $A_{k,n}$ the remaining $n$-dependent asymptotic term. The source gives

$$\alpha_{k,n}=n+\operatorname{frac}\Bigl(\frac{k+4}{8}\Bigr)+\frac{k}{8(2k+1)}+A_{k,n}$$

and

$$\alpha_{k,n}-\nu_k(n/\kappa)=\frac{2k+9}{16}-\frac{1}{16(2k+1)}+A_{k,n}.$$

Exactness conjecture for the median asymptotics. The terms in these two formulas that depend only on $k$ are exact.

The $n$-dependent terms are explicitly described as numerical fits, while the $k$-only terms are conjectured to be exact.