Nadjafi-Arani et al.'s Szeged–Wiener inequality for non-complete blocks

Let $G$ be a connected graph and let $B_1,\ldots,B_k$ be all its non-complete blocks, of respective orders $n_1,\ldots,n_k$. Define the Szeged–Wiener difference by $\eta(G)=\operatorname{Sz}(G)-W(G)$. Nadjafi-Arani et al.'s conjecture. One has

$$\eta(G)\ge \sum_{i=1}^k(2n_i-6).$$

The paper proves this conjecture as a consequence of its theorem for $2$-connected non-complete graphs, so the proposed inequality is solved.