The first counterexamples in dimension greater than 3 were from Ishii and Kollarin 2002. In 2011 we proved the conjecture for surfaces in a joint work with Javier Fernández de Bobadilla. In the summer of 2012 T. de Fernex gave counteresamples in dimension 3 and afterwards, J. Kollar found more counterexamples in these dimensions. Recently, R. Docampo and T. de Fernex proved a result concerning the space of arcs and the terminal minimal models of the singularity as a partial result for the problem in higher dimension.
In this talk, I will give and introduction to the problem and details of the proof of the conjecture for the case of normal surfaces.