Complete-graph extremality conjecture for mean subtree order

Let $G$ be a graph with $n$ vertices, and let $\mu(G)$ denote its mean subtree order. Let $K_n$ be the complete graph on $n$ vertices.

Complete-graph extremality conjecture. For every graph $G$ with $n$ vertices,

$$\mu(G)\le\mu(K_n).$$

This is the maximum part of the main extremal problem for mean subtree order. The source identifies it as open.