The flat-section form of Gamma-conjecture I

Let $X$ be a Fano manifold and let $\mathcal{A}$ be the vector space of flat sections $s(z)$ over the positive real axis such that $e^{T/z}s(z)$ has at most polynomial growth as $z\to+0$, where $T$ is the maximal norm of the eigenvalues of $c_1(X)\star_0$. Let $\mathfrak{s}(\mathcal{O})$ denote the flat section associated with the structure sheaf. The flat-section form of Gamma-conjecture I. The space $\mathcal{A}$ is generated by $\mathfrak{s}(\mathcal{O})|_{\tau=0}$. Assuming Conjecture O, the source explains that $\mathcal{A}$ is one-dimensional and its sections have limiting $T$-eigenvectors; this formulation is presented as the original Gamma-conjecture I.