A-model conifold eigenvalue conjecture for Fano manifolds

For a Fano manifold $X$, let $G_X(t)=\sum_{n=0}^{\infty}a_nt^n/n!$ be its quantum period and define the A-model conifold value by

$$T_{\mathrm{A,con}}:=\limsup_{n\to\infty}|a_n|^{1/n}.$$

Let $\hat c_1$ denote quantum multiplication by $c_1(X)$ on the even cohomology. A-model conifold eigenvalue conjecture. The number $T_{\mathrm{A,con}}$ is an eigenvalue of $\hat c_1$. The claim is proved in several classes, including toric Fano manifolds and Fano threefolds, but remains conjectural for arbitrary Fano manifolds.