Maximizing the Mostar index for bipartite graphs and split graphs

Do\v{s}li\'{c} et al.~defined the Mostar index of a graph $G$ as $\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. Contributing to conjectures posed by Do\v{s}li\'{c} et al., we show that the Mostar index of bipartite graphs of order $n$ is at most $\frac{\sqrt{3}}{18}n^3$, and that the Mostar index of split graphs of order $n$ is at most $\frac{4}{27}n^3$.