Wiener–eccentricity monotonicity conjecture under edge contraction

Let $G$ be a graph with at least three vertices, and let $e$ be an edge of $G$. Write $G.e$ for the graph obtained by contracting $e$. The Wiener index is

$$W(G)=\sum_{\{u,v\}\subset V(G)}d_G(u,v),$$

and the eccentricity of $G$ is

$$\varepsilon(G)=\sum_{u\in V(G)}\varepsilon_G(u).$$

Wiener–eccentricity monotonicity conjecture. For every such edge $e$,

$$W(G.e)-\varepsilon(G.e)\leq W(G)-\varepsilon(G).$$

The conjecture asserts that contracting an edge cannot increase the difference between the Wiener index and total eccentricity. The source notes that the claim was proved when $e$ is a bridge; its general status is not specified in the supplied text.