Do homological eigenvalues exhaust the spectrum for p in (1,∞)?

Determine whether, for p in (1,∞), every p-Laplacian eigenvalue is a homological critical value of the p-Rayleigh quotient; equivalently, decide whether non-homological eigenvalues can occur in this regime.

Background

Homological eigenvalues are defined via changes in the homology of sublevel sets of the Rayleigh quotient. In finite dimensions, homological critical values are classical critical values, and for p=1 there are known examples of eigenvalues that are not homological.

Whether the homological spectrum matches the full spectrum for p in (1,∞) remains unsettled. An affirmative answer would strongly constrain the structure of nonlinear spectra; a negative answer would exhibit fundamentally non-homological eigenvalues in the nonlinear discrete setting.