  In this paper, we study left invariant conic Finsler metrics on the
2-dimensional non-Abelian Lie group $G$ with nowhere vanishing spray vector
fields, and classify those satisfying the constant curvature condition, the
Landsberg condition or the Berwald condition respectively. We prove that any
left invariant conic Landsberg metric on $G$ must be Berwald. This discovery
enable us to propose a homogeneous conic Landsberg Conjecture, which guesses
that every homogeneous conic Landsberg metric is Berwald, and prove the
2-dimensional case for it.
