Gutman–Mizoguchi real-rootedness conjecture for beta-polynomials

Let $C$ be a circuit contained in a graph $G$. The matching polynomial is denoted by $\alpha(G,x)$, and the two circuit characteristic, or beta-polynomials, considered here are

$$\beta(G,C,x)=\alpha(G,x)-2\alpha(G-C,x)$$

and

$$\beta(G,C,x)=\alpha(G,x)+2\alpha(G-C,x).$$

Here $G-C$ is obtained by deleting the vertices of $C$. Gutman–Mizoguchi conjecture. For any circuit $C$ contained in any graph $G$, both beta-polynomials are real, meaning that all their zeros are real numbers. The conjecture was motivated by proofs for several classes of graphs and extensive computer searches that found no beta-polynomial with a non-real zero; its general case remains open.