The path and clique extremality conjecture for mean subtree order

Let $G$ be a connected graph of order $n$, and let $\mu(G)$ denote its mean subtree order. Write $P_n$ for the path and $K_n$ for the clique on $n$ vertices.

Mean subtree order extremality conjecture.

$$\mu(P_n) \le \mu(G) \quad\text{with equality if and only if }G=P_n,$$

and

$$\mu(G) \le \mu(K_n) \quad\text{with equality if and only if }G=K_n.$$

The minimum assertion is proved in the paper, while the maximum assertion remains open; thus the full conjecture is not resolved.