Asymptotic results for the Poisson distribution of order k

The Poisson distribution of order $k$ is a special case of a compound Poisson distribution. Its mean and variance are known, but results for its median and mode are difficult to obtain, although a few cases have been solved and upper/lower bounds for the mode have been established. This note points out that asymptotic results, for both the median and mode, exhibit simple patterns. The calculations are numerical, hence the results are presented as conjectures. The purpose of this note is to discern patterns for the median and mode, including expressions for exact limits and rates of convergence and (possibly sharp) upper/lower bounds, in a sense to be made precise in the text. The derivation of proofs of the results is left for future work.