A graph is vertex-transitive if its automorphism group acts transitively upon its vertices.

A Hamiltonian path is a path that visits each vertex exactly once.

Lovász conjecture: Every finite connected vertex-transitive graph contains a Hamiltonian path.

This conjecture is open since almost 50 years.

A connected graph is $2$-connected if it remains connected whenever any vertex is removed.

**Question**: Is Lovász conjecture known in the $2$-connected case (or $k$-connected for $k$ large enough)?