Matthews–Sumner conjecture on Hamilton cycles in 4-connected claw-free graphs

Let $G$ be a graph. It is claw-free if it has no induced subgraph isomorphic to a claw, and $G$ is 4-connected if deleting fewer than four vertices does not disconnect it. Matthews–Sumner conjecture. Every $4$-connected, claw-free graph has a Hamiltonian cycle.

This is a central conjecture on Hamiltonicity in sparse graphs. The supplied paper describes it as a famous open problem and discusses it as motivation for a bipartite analogue.