Connectedness of Lakshmibai-Seshadri path crystals for hyperbolic Kac-Moody algebras of rank 2

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$. We give a necessary and sufficient condition for the crystal graph of the Lakshmibai-Seshadri path crystal $\mathbb{B}(\lambda )$ to be connected for an arbitrary integral weight $\lambda$.