Gamma Conjecture I for Fano manifolds

Let $X$ be a Fano manifold. Denote by $c_1(X)\star_0$ quantum multiplication by the first Chern class at the origin, and suppose that it has a simple rightmost eigenvalue. Let $\hat{\Gamma}_X$ be the Gamma class of $X$, and let $A_X$ be the principal asymptotic class of $X$. Gamma Conjecture I. The Gamma class gives the principal asymptotic class, namely

$$\hat{\Gamma}_X \in \mathbb{C} \cdot A_X.$$

This conjecture identifies the Gamma class, up to a non-zero constant multiple, with the principal asymptotic class whenever the latter is defined under the simple-rightmost-eigenvalue hypothesis. It relates characteristic-class data to the asymptotic behaviour of the quantum $D$-module; the supplied text gives no resolution status.