Gutman–Furtula revised conjecture on trees with minimal ABC index

Let $G$ be a tree with minimal atom-bond connectivity (ABC) index among all trees of size $n$. The trees $T_0,T_1,T_2,T_3,T_4,T_5,T_6$ are the structures depicted in Figure~. Gutman–Furtula's revised conjecture. If $n$ satisfies one of the following congruence conditions, then $G$ has the corresponding structure:

$$\begin{array}{ll} n \equiv 0 \pmod{7},\ n \geq 175 &\Longrightarrow G\text{ has structure }T_0,\\ n \equiv 1 \pmod{7},\ n \geq 64 &\Longrightarrow G\text{ has structure }T_1,\\ n \equiv 2 \pmod{7},\ n \geq 1185 &\Longrightarrow G\text{ has structure }T_2,\\ n \equiv 3 \pmod{7},\ n \geq 80 &\Longrightarrow G\text{ has structure }T_3,\\ n \equiv 4 \pmod{7},\ n \geq 312 &\Longrightarrow G\text{ has structure }T_4,\\ n \equiv 5 \pmod{7},\ n \geq 117 &\Longrightarrow G\text{ has structure }T_5,\\ n \equiv 6 \pmod{7},\ n \geq 62 &\Longrightarrow G\text{ has structure }T_6. \end{array}$$

This is a revised version of the conjecture of Gutman and Furtula after counterexamples to earlier formulations based on a central-vertex structure. The paper reports computational evidence and notes that determining the extent of the violation of the earlier conjectures remains open.