Gamma conjecture for asymptotics of mirror Lagrangian periods

Let $X$ be a Calabi--Yau manifold equipped with a symplectic form $\omega$, and let $\{Z_t\}_{t\in \Delta^\ast}$ be a family of Calabi--Yau manifolds parametrized by $t$ in a small punctured disc $\Delta^\ast$ that corresponds to $(X,\omega)$ under mirror symmetry. Let $\Omega_t$ be a suitable holomorphic volume form on $Z_t$, let $t$ be a suitable coordinate, and let $C_t\subset Z_t$ be a Lagrangian cycle mirror to a coherent sheaf $E$ on $X$. Gamma conjecture. As $t\to 0$ in a fixed angular sector, one should have

$$\int_{C_t} \Omega_t=\int_X t^{-\omega}\cdot \widehat{\Gamma}_X\cdot (2\pi\sqrt{-1})^{\deg/2}\operatorname{ch}(E)+O(t^\epsilon),$$

where $\epsilon>0$ is some constant. This conjectural asymptotic formula relates periods of Lagrangian cycles on the mirror family to the Gamma class and Chern character on the Calabi--Yau manifold, and is motivated by the mirror-symmetric correspondence between tropical or ordinary cycles and the Gamma conjecture. Its general validity remains open.