Berry-Esseen bound for binomial distribution

The paper "On The Bound of Proximity of the Binomial Distribution to the Normal One" - Nagaev, Chebotarev (2010) has improvements to $C$ specifically for the Binomial Distribution.

Theorem 2 on page 3 gathers together the results in the paper to show that $C$ can be taken to be .4215 in general. The paper notes that an Esseen (1956) paper demonstrates that the bound has to be at least $$\frac{\sqrt{10} + 3}{6\sqrt{2\pi}} \approx .409732$$ so they have found a relatively close bound.

However, there are some better bounds stated in the paper for specific situations. When $n \leq 2000$ and $.02 \leq p \leq .5$ the bound can be taken to be $.4096$. When $0 < p < .02$ it can be taken to be $.369$.