Abstract: We discuss the problem of desingularizing nearly Kähler and nearly G_2 manifolds with conical singularities using asymptotically conical Calabi--Yau 3-folds or G_2 manifolds. The first obstruction to such a desingularization turns out to be the exactness of certain forms on the AC manifolds. A rigidity result shows that below the rate threshold -3 for the AC manifold in question, this can only happen for Euclidean spaces.