The strengthened Szeged–Wiener lower bound for 2-connected graphs

Let $G$ be a $2$-connected graph of order $n\ge 10$. Let $K_n$ denote the complete graph, and let $K_n^2$ and $K_n^{n-2}$ be the graphs obtained from $K_{n-1}$ by adding one vertex adjacent to respectively $2$ or $n-2$ of the original vertices. Define $\eta(G)=\operatorname{Sz}(G)-W(G)$. Strengthened Szeged–Wiener conjecture. If $G$ is not isomorphic to $K_n$, $K_n^2$, or $K_n^{n-2}$, then

$$\eta(G)\ge 2n.$$

The conjecture is motivated by computer searches finding graphs with $\eta(G)<2n$ only up to order $9$, with none on $10$ vertices. The source does not provide a proof or resolution, so the conjecture remains open.