Gross–Siebert's period-integral conjecture for toric Calabi–Yau mirrors

Let $\check{X}_t$ be the mirror family of a toric Calabi–Yau variety, let $\check{\Omega}_{\phi(q)}$ be its holomorphic volume form evaluated at the SYZ-map parameter $\phi(q)$, and let $q=(q_1,\ldots,q_r)$ be the complexified Kähler parameters. Suppose that $\phi$ is defined using the generating functions $1+\delta_i(q)$ of the genus-zero open Gromov–Witten invariants $n_{\beta_i+\alpha}$. Gross–Siebert's conjecture. There exist integral cycles $\Gamma_1,\ldots,\Gamma_r$, forming part of an integral basis of $H_n(\check{X}_t;\mathbb{Z})$, such that

$$q_a=\exp\left(-\int_{\Gamma_a}\check{\Omega}_{\phi(q)}\right),\qquad a=1,\ldots,r.$$

Equivalently, the SYZ map is inverse to a mirror map. The conjecture predicts that the SYZ construction naturally produces canonical coordinates, relating open disk counts to period integrals on the mirror family.