Uniform oscillation bound on arbitrary membrane subdomains

Determine whether, for the vibrating membrane equation on a rectangular domain, every open subset that does not contain the intersection of a strip parallel to a side with the rectangle nevertheless admits a uniform time bound, independent of the solution, within which every nonzero solution takes both positive and negative values.

Background

For vibrating membranes, the paper establishes that oscillation times in an open subset are uniformly bounded when the subset contains the intersection of a strip parallel to one side with the rectangular domain. The authors then state that the existence of such a bound is unresolved for all other subsets. This problem asks for a characterization of those observation regions that possess a solution-independent oscillation-time bound.