All genus open mirror symmetry for the projective line

We prove that the Chekhov-Eynard-Orantin recursion on the mirror curve of ${\mathbb{P}}^1$ encodes all genus equivariant open Gromov-Witten invariants of $({\mathbb{P}}^1,\mathbb{R}{\mathbb{P}}^1)$. This result can be viewed as an all genus equivariant open mirror symmetry for $({\mathbb{P}}^1,\mathbb{R}{\mathbb{P}}^1)$.