Weak Gamma conjecture I

Let $X$ be a Fano manifold. Let $S(z)$ be the quantum differential-equation fundamental solution, $\mu$ the grading operator, $c_1$ act by quantum multiplication, and $\widehat\Gamma_X$ be the Gamma class. Weak Gamma conjecture I. There exist classes $\alpha_0,\alpha_1,\alpha_2,\ldots\in H^\bullet(X)$, with $\alpha_0$ a nonzero eigenvector of $\hat c_1$ of eigenvalue $T>0$, such that

$$e^{T/z}S(z)z^{-\mu}z^{c_1}\widehat\Gamma_X\sim\alpha_0+\alpha_1z+\alpha_2z^2+\cdots$$

as $z\to0$ in the sector $|\arg z|<\pi/2+\epsilon$ for some $\epsilon>0$. This is a weaker asymptotic version of Gamma conjecture I; the source notes verification for all toric Fano manifolds but does not claim a general proof.