  The atom-bond connectivity (ABC) index is one of the most investigated
degree-based molecular structure descriptors with a variety of chemical
applications. It is known that among all connected graphs, the trees minimize
the ABC index. However, a full characterization of trees with a minimal ABC
index is still an open problem. By now, one of the proved properties is that a
tree with a minimal ABC index may have, at most, one pendent path of length
$3$, with the conjecture that it cannot be a case if the order of a tree is
larger than $1178$. Here, we provide an affirmative answer of a strengthened
version of that conjecture, showing that a tree with minimal ABC index cannot
contain a pendent path of length $3$ if its order is larger than $415$.
