Bounding the distance among longest paths in a connected graph

It is easy to see that in a connected graph any 2 longest paths have a vertex in common. For k≥7, Skupień in 1966 obtained a connected graph in which some k longest paths have no common vertex, but every k−1 longest paths have a common vertex. It is not known whether every 3 longest paths in a connected graph have a common vertex and similarly for 4, 5, and 6 longest path. Fujita et al. in 2015 give an upper bound on distance among 3 longest paths in a connected graph. In this paper we give a similar upper bound on distance between 4 longest paths and also for k longest paths, in general.