Modified Gamma conjecture I, strong form

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. A flat section associated with $\widehat\Gamma_X$ is said to have moderate growth after multiplication by $e^{T/z}$ if

$$e^{T/z}S(z)z^{-\mu}z^{c_1}\widehat\Gamma_X$$

has moderate growth as $z\to0$ in the sector $|\arg z|<\pi/2+\epsilon$. Modified Gamma conjecture I, strong form. There exist $T\in\mathbb C$ and $\epsilon>0$ such that this moderate-growth condition holds, and $T=T_{\mathrm{A,con}}$. This strong formulation is verified for toric Fano manifolds in the paper and is proposed more generally as a modification of Gamma conjecture I.