  Thomassen's chord conjecture from 1976 states that every longest cycle in a
$3$-connected graph has a chord. This is one of the most important unsolved
problems in graph theory. We pose a new conjecture which implies Thomassen's
conjecture. It involves bound vertices in a longest path between two vertices
in a $k$-connected graph. We also give supporting evidence and analyze a
special case. The purpose of making this new conjecture is to explore the
surroundings of Thomassen's conjecture.
