Aouchiche–Hansen–Zheng conjecture on the Randić index and matching number

Let $G$ be a connected graph on $n$ vertices, where $n\ge 3$. Write $R(G)=\sum_{uv\in E(G)}(d(u)d(v))^{-1/2}$ for its Randić index and let $\alpha'(G)$ denote its matching number. Define

$$p=\left\lfloor\frac{n+4}{7}\right\rfloor,\qquad q=\left\lfloor\frac{6n+2}{7}\right\rfloor.$$

Aouchiche–Hansen–Zheng conjecture. One has

$$R(G)-\alpha'(G)\le\sqrt{\left\lfloor\frac{n+4}{7}\right\rfloor\left\lfloor\frac{6n+2}{7}\right\rfloor}-\left\lfloor\frac{n+4}{7}\right\rfloor,$$

with equality if and only if $G=K_{p,q}$.

The conjecture proposes the extremal connected graph for the difference between the Randić index and matching number. It was disproved by the existence of a counterexample; the corresponding restricted extremal problem for subcubic graphs was solved by Du, Hu, and collaborators.