The complete multipartite extremal conjecture for the maximum ABC index

Let $G$ be an $n$-vertex connected graph with chromatic number $\chi\geq 3$. Let $T_{n,\chi}$ denote the complete $\chi$-partite graph of order $n$ whose part sizes differ by at most one. The quantity ${\rm ABC}(G)$ denotes the atom-bond connectivity index.

The complete multipartite extremal conjecture.

$${\rm ABC}(G)\leq {\rm ABC}(T_{n,\chi}),$$

with equality if and only if $G\cong T_{n,\chi}$.

This conjecture proposes that, among connected graphs with fixed order and chromatic number at least $3$, the balanced complete multipartite graph maximizes the ABC index. The supplied text gives no resolution of the conjecture.