3 problems

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.

Li et al.'s conjecture. For every pair of vertices $a,b$ of a connected line graph of a hypergraph of rank at most $3$, there is a maximal $(a,b)$-path which is a Tutte path.

Jackson's conjecture. Every $2$-connected claw-free graph has a Tutte cycle.