  In this paper we prove Gamma Conjecture $1$ for twistor bundles of hyperbolic
$6$ manifolds, which are monotone symplectic manifolds which admit no K\"ahler
structure. The proof involves a direct computation of the $J$-function, and a
version of Laplace's method for estimating power series (as opposed to
integrals). This method allows us to rephrase Gamma Conjecture $1$ in certain
situations to an Ap\'ery-like discrete limit. We use this to give a simple
proof of Gamma Conjecture $1$ for projective spaces. Additionally we show that
the quantum connections of the twistor bundles we consider have unramified
exponential type.
