We show that the $L_1$ norm of the difference between the standard normal distribution and the distribution of the standardized sum of $n$ independent random variables is less than 72 $R_n$, where $R_n$ is a sum of standardized "inside" third and "outside" second moments. We conjecture that 72 can be replaced by 36 or even less. We also prove a similar result for $m$-dependent random variables, but no constant is specified.
"On an $L_p$ Version of the Berry-Esseen Theorem for Independent and $m$- Dependent Variables." Ann. Probab. 1 (3) 497 - 503, June, 1973. https://doi.org/10.1214/aop/1176996944