In 1989, M. Anderson conjectured that the Gromov-Hausdorff limit of a sequence of complete manifolds with bounded Ricci curvature has only singularities of codimension larger than four. This was proven in 2015 by J. Cheeger and A. Naber. In this talk, I will present a joint work with G. Carron and D. Tewodrose in which we prove an analogue result under a Kato condition on the Ricci curvature.