Longest cycles through a specified edge in 3-connected graphs

Establish that, for every 3-connected graph and every specified edge, each longest cycle containing that edge has a chord.

Background

This conjecture is a localized strengthening of the general question about chords in longest cycles. Instead of considering all longest cycles in a 3-connected graph, it fixes an edge and considers cycles containing that edge.

The authors prove the conjecture in several special settings, including 3-connected graphs of minimum degree at least four for the existence of some suitable longest cycle, and certain planar graph cases where every longest cycle through a specified small set has a chord. The full assertion that every longest cycle containing the specified edge has a chord is not resolved by those results.