  We show how an inaccurate determination of experimental uncertainty
correlations in high-precision LHC measurements may undermine the reliability
of the associated $\chi^2$. We formulate the problem rigorously, and devise a
regularisation procedure that increases the stability of the $\chi^2$ by
altering the covariance matrix of the measurement as little as possible. We
apply the procedure to the NNPDF4.0 global analysis of parton distribution
functions that utilises a large amount of LHC measurements. We find that the
regularised $\chi^2$ of the NNPDF4.0 determination is lowered by about
$3\sigma$, without significantly altering the resulting PDFs upon refitting.
