Tutte-path conjecture for connected claw-free graphs

A graph is claw-free if it has no induced subgraph isomorphic to $K_{1,3}$. For vertices $a$ and $b$, an $(a,b)$-path is a path with endpoints $a$ and $b$. A Tutte path is an $(a,b)$-path whose off-path components have at most three neighbours on the path, and a path is maximal if it cannot be extended while preserving the relevant path conditions.

Claw-free Tutte-path conjecture. For every pair of vertices $a,b$ of a connected claw-free graph, there is a maximal $(a,b)$-path which is a Tutte path.

The paper presents this as a seemingly stronger reformulation of Jackson's conjecture and proves that it is equivalent to the same open problem. Thus the assertion remains open despite the equivalence with the other formulations.