The Graovac-Ghorbani index upper-bound conjecture for bicyclic graphs

Let $\mathcal{B}'_n$ be the family of all bicyclic graphs on $n$ vertices, and let $G\in\mathcal{B}'_n$ have order $n\geq 8$. The Graovac-Ghorbani atom-bond connectivity index is denoted by $ABC_{GG}(G)$. Let $H$ be the graph obtained by adding $n-4$ pendant vertices to one degree-$3$ vertex of $K_4$ with one edge removed. Upper-bound conjecture.

$$ABC_{GG}(G)\leq (n-4)\sqrt{\frac{n-2}{n-1}}+\sqrt{\frac{n-4}{n-3}}+2\sqrt{\frac{n-3}{n-2}}+\frac{\sqrt{2}}{2}.$$

Equality holds if and only if $G$ is isomorphic to $H$. Computational experiments suggest that this graph gives the maximum of the index among all bicyclic graphs for $n\geq 8$.