How small can the hole be in Neumann counterexamples?

Ascertain how small the inner boundary component can be in a bounded doubly-connected planar domain that admits a second Neumann eigenfunction with a closed nodal line fully contained in the interior; in particular, determine whether such counterexamples exist with arbitrarily small holes or even with the hole reduced to a fracture (slit).

Background

The paper proves existence of doubly-connected planar domains whose second Neumann eigenfunction has a closed interior nodal line. The authors then ask how far this can be pushed in terms of the size of the hole.

They remark that, unlike in the Dirichlet setting, their present techniques do not show that the hole can be reduced to a fracture in the Neumann case, motivating a precise quantitative question.