Hansen's Szeged–Wiener index conjecture for bipartite graphs

Let $G$ be a finite, simple, connected bipartite graph with $n \geq 4$ vertices and $m \geq n$ edges. Its Wiener index is

$$W(G)=\sum_{\{u,v\}\subseteq V(G)}d(u,v),$$

and its Szeged index is

$$Sz(G)=\sum_{e=uv\in E(G)}n_u(e)n_v(e),$$

where $n_u(e)$ and $n_v(e)$ count the vertices closer to $u$ and $v$, respectively, for the edge $e=uv$. Hansen's conjecture.

$$Sz(G)-W(G)\geq 4n-8.$$

The bound is best possible: equality is attained by a graph formed from a $4$-cycle $C_4$ and a tree $T$ on $n-3$ vertices sharing one vertex. The conjecture concerns a lower bound relating two classical graph indices and, in the stated source, no resolution is supplied.