Hansen–Li–Liu conjecture on the revised Szeged–Wiener difference for nonbipartite graphs

Let $G$ be a connected graph with $n \geq 4$ vertices, $m \geq n$ edges, and an odd cycle. The Hansen–Li–Liu conjecture. asserts that

$$Sz^*(G)-W(G) \geq \frac{n^2+4n-6}{4}.$$

Moreover, the bound should be best possible, attained by the graph composed of a cycle on $3$ vertices, $C_3$, and a tree $T$ on $n-3$ vertices sharing a single vertex. Here $W(G)$ is the Wiener index and $Sz^*(G)$ is the revised Szeged index. The conjecture proposes a sharp lower bound for their difference among connected nonbipartite graphs with at least as many edges as vertices.