We study the thickening problem on a 2-dimensional Stein variety X with isolated irreducible singularities, i.e. the problem whether the assumption that a compact set K is contained in the interior of another compact set L implies that the same inclusion holds for their holomorphic hulls. The problem is still open except for a positive answer in the special case of quotient singularities. The main result of the present article is the partial result that the holomorphic hull L is has a directional density property at every singular point contained in the hull of K. The proof is based on removability results on pseudoconcave closed sets, which may be of some independent interest.