  Payne conjectured in 1967 that the nodal line of the second Dirichlet
eigenfunction must touch the boundary of the domain. In their 1997 breakthrough
paper, Hoffmann-Ostenhof, Hoffmann-Ostenhof and Nadirashvili proved this to be
false by constructing a counterexample in the plane with many holes and raised
the question of the minimum number of holes a counterexample can have. In this
paper we prove it is at most 6.
