  Proportional choosability is a list analogue of equitable coloring that was
introduced in 2019. The smallest $k$ for which a graph $G$ is proportionally
$k$-choosable is the proportional choice number of $G$, and it is denoted
$\chi_{pc}(G)$. In the first ever paper on proportional choosability, it was
shown that when $2 \leq n \leq m$, $ \max\{ n + 1, 1 + \lceil m / 2 \rceil\}
\leq \chi_{pc}(K_{n,m}) \leq n + m - 1$. In this note we improve on this result
by showing that $ \max\{ n + 1, \lceil n / 2 \rceil + \lceil m / 2 \rceil\}
\leq \chi_{pc}(K_{n,m}) \leq n + m -1- \lfloor m/3 \rfloor$. In the process, we
prove some new lower bounds on the proportional choice number of complete
multipartite graphs. We also present several interesting open questions.
