  The search for non-Gaussian signatures in the Cosmic Microwave Background
(CMB) is crucial for understanding the physics of the early Universe. Given the
possibility of non-Gaussian fluctuations in the CMB, a recent revision to the
standard $\Lambda$-Cold Dark Matter ($\Lambda$CDM) model has been proposed,
dubbed "Super-$\Lambda$CDM". This model introduces additional free parameters
to account for the potential effects of a trispectrum in the primordial
fluctuations. In this study, we explore the impact of the Super-$\Lambda$CDM
model on current constraints on neutrino physics. In agreement with previous
research, our analysis reveals that for most of the datasets, the
Super-$\Lambda$CDM parameter $A_0$ significantly deviates from zero at over a
$95\%$ confidence level. We then demonstrate that this signal might influence
current constraints in the neutrino sector. Specifically, we find that the
current constraints on neutrino masses may be relaxed by over a factor of two
within the Super-$\Lambda$CDM framework, thanks to the correlation present with
$A_0$. Consequently, locking $A_0=0$ might introduce a bias, leading to overly
stringent constraints on the total neutrino mass.
