Enumerative meaning of mirror maps for toric Calabi-Yau manifolds

We prove that the inverse of a mirror map for a toric Calabi-Yau manifold of the form KY, where Y is a compact toric Fano manifold, can be expressed in terms of generating functions of genus 0 open Gromov-Witten invariants defined by Fukaya-Oh-Ohta-Ono (2010)  [15]. Such a relation between mirror maps and disk counting invariants was first conjectured by Gross and Siebert (2011)  [24, Conjecture 0.2 and Remark 5.1] as part of their program, and was later formulated in terms of Fukaya-Oh-Ohta-Ono's invariants in the toric Calabi-Yau case in Chan et al. (2012)  [8, Conjecture 1.1].