  Accurate inference of cosmology from weak lensing shear requires an accurate
shear power spectrum covariance matrix. Here, we investigate this accuracy
requirement and quantify the relative importance of the Gaussian (G),
super-sample covariance (SSC) and connected non-Gaussian (cNG) contributions to
the covariance. Specifically, we forecast cosmological parameter constraints
for future wide-field surveys and study how different covariance matrix
components affect parameter bounds. Our main result is that the cNG term
represents only a small and potentially negligible contribution to statistical
parameter errors: the errors obtained using the G+SSC subset are within
$\lesssim 5\%$ of those obtained with the full G+SSC+cNG matrix for a
Euclid-like survey. This result also holds for the shear two-point correlation
function, variations in survey specifications and for different analytical
prescriptions of the cNG term. The cNG term is that which is often tackled
using numerically expensive ensembles of survey realizations. Our results
suggest however that the accuracy of analytical or approximate numerical
methods to compute the cNG term is likely to be sufficient for cosmic shear
inference from the next generation of surveys.
