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In the paper arXiv:1411.4887 [math.AP] it is shown that the set of Riemannianmetrics which do not admit global limiting Carleman weights is open and dense,by studying the conformally invariant Weyl and Cotton tensors. In the paperarXiv:1011.2507 [math.DG] it is shown that the set of Riemannian metrics whichdo not admit local limiting Carleman weights at any point is residual, showingthat it contains the set of metrics for which there are no local conformaldiffeomorphisms between any distinct open subsets. This paper is a continuationof arXiv:1411.4887 [math.AP] in order to prove that the set of Riemannianmetrics which do not admit local limiting Carleman weights \emph{at any point}is open and dense.

On the set of metrics without local limiting Carleman weights