It has been conjectured that the minimum and maximum of the mean subtree order among connected graphs of order $n$ are attained by the path $P_n$ and clique $K_n$, respectively. Extending ideas due to Haslegrave and Vince, we confirm that the minimum is indeed attained by $P_n$. On the other hand, we discuss different approaches (both promising and flawed) that could lead to a proof of the extremality of $K_n$.