The line-graph inertia conjecture

Let $G$ be a connected graph, and let $L(G)$ denote its line graph. Let $n^+(L(G))$ and $n^-(L(G))$ be the numbers of positive and negative eigenvalues of the adjacency matrix of $L(G)$, and let $s(L(G))$ be its signature. Line-graph inertia conjecture.

$$n^+(L(G))\leq n^-(L(G))+1.$$

Equivalently, $s(L(G))\leq 1$. The paper proves a weaker upper bound for line graphs and presents computational evidence for this sharper conjecture, which remains open.