Connected-graph contraction conjecture for mean subtree order

Let $G$ be a connected graph and let $e\in E(G)$. Write $G/e$ for the graph obtained by contracting $e$, and let $\mu$ denote mean subtree order.

Connected-graph contraction conjecture. Contracting any edge should reduce the mean subtree order by at least $\frac13$, with equality only for a path:

$$\mu(G)-\mu(G/e)\ge\frac13,$$

with equality if and only if $G$ is a path. The source presents this as a proposed generalisation of the proved tree case and says that proving it would imply the path-minimum result.