Aouchiche's Randić index–diameter bounds conjecture

Let $G$ be a graph with $n$ vertices. Write $R=R(G)$ for its Randić index and $d$ for its diameter. Aouchiche's conjecture. The two bounds

$$R-d\geq \sqrt 2-\frac{n+1}{2},$$

and

$$\frac{R}{d}\geq \frac{n-3+2\sqrt 2}{2n-2}$$

should hold. These inequalities connect the Randić index with graph diameter. The supplied text subsequently reports that both bounds were verified for all graphs, so the conjecture is resolved.