Ma et al.'s signature bounds conjecture for simple graphs

Let $G$ be a simple graph with adjacency matrix $A(G)$. Its signature is $s(G)=p(G)-n(G)$, where $p(G)$ and $n(G)$ are respectively the numbers of positive and negative eigenvalues of $A(G)$. Let $c_3(G)$ and $c_5(G)$ denote respectively the numbers of cycles in $G$ whose lengths are $4k+3$ and $4k+5$ for some integers $k\geq 0$. Ma et al.'s signature bounds conjecture. The inequality

$$-c_3(G)\leq s(G)\leq c_5(G)$$

possibly holds for every simple graph $G$. Ma et al. proved the inequality for trees, unicyclic graphs, and bicyclic graphs. This paper proves it for line graphs and power trees, while the general case is not established here.