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The next generation of ground-based CMB experiments aim to measuretemperature and polarization fluctuations up to $\ell_{\rm max} \approx 5000$over half of the sky. Combined with Planck data on large scales, this willprovide improved constraints on primordial non-Gaussianity. However, theimpressive resolution of these experiments will come at a price. Besides signalconfusion from galactic foregrounds, extra-galactic foregrounds and late-timegravitational effects, gravitational lensing will introduce largenon-Gaussianity that can become the leading contribution to the bispectrumcovariance through the connected 4-point function. Here, we compute this effectanalytically for the first time on the full sky for both temperature andpolarization. We compare our analytical results with those obtained directlyfrom map-based simulations of the CMB sky for several levels of instrumentalnoise. Of the standard shapes considered in the literature, the local shape ismost affected, resulting in a 35\% increase of the estimator standard deviationfor an experiment like the Simons Observatory (SO) and a 110\% increase for acosmic-variance limited experiment, including both temperature and polarizationmodes up to $\ell_{\rm max} = 3800$. Because of the nature of the lensing4-point function, the impact on other shapes is reduced while still nonnegligible for the orthogonal shape. Two possible avenues to reduce thenon-Gaussian contribution to the covariance are proposed. First bymarginalizing over lensing contributions, such as the ISW-lensing 3pt functionin temperature, and second by delensing the CMB. We show the latter method canremove almost all extra covariance, reducing the effect to below $<$5\% forlocal bispectra. At the same time, delensing would remove signal biases fromsecondaries induced by lensing, such as ISW-lensing.

Minimizing gravitational lensing contributions to the primordial bispectrum covariance