Gallai’s common-vertex problem for three longest paths

Determine whether every triple of longest paths in a finite connected graph has a common vertex.

Background

This problem asks for a Helly-type intersection property for longest paths: pairwise or individual maximality is not enough to make the existence of a common vertex immediate.

The question is attributed to Gallai and is cited from the British Combinatorial Conference problem list. The paper does not resolve it.