  The Eigenstate Thermalization Hypothesis makes a prediction for the
statistical distribution of matrix elements of simple operators in energy
eigenstates of chaotic quantum systems. As a leading approximation,
off-diagonal matrix elements are described by Gaussian random variables but
higher-point correlation functions enforce non-Gaussian corrections which are
further exponentially suppressed in the entropy. In this paper, we investigate
non-Gaussian corrections to the statistical distribution of heavy-heavy-heavy
OPE coefficients in chaotic two-dimensional conformal field theories. Using the
Virasoro crossing kernels, we provide asymptotic formulas involving arbitrary
numbers of OPE coefficients from modular invariance on genus-$g$ surfaces. We
find that the non-Gaussianities are further exponentially suppressed in the
entropy, much like the ETH. We discuss the implication of these results for
products of CFT partition functions in gravity and Euclidean wormholes. Our
results suggest that there are new connected wormhole geometries that dominate
over the genus-two wormhole.
