Classical spectral problems, such as the Dirichlet and Neumann problems, focus on the analysis of eigenvalues and eigenfunctions, with applications to heat conduction, sound propagation, and vibrational modes in geometric domains. Other well-known problems are the Steklov and biharmonic Steklov problems with Dirichlet or Neumann boundary conditions. Kuttler and Sigillito established fundamental inequalities relating the eigenvalues of these problems in planar domains. These results were later extended to the scalar case on Riemannian manifolds by Hassannezhad and Siffert. In this talk, we propose a generalization of the biharmonic Steklov problem with Neumann boundary conditions to the setting of differential forms, which will help us extend the spectral inequalities of Kuttler and Sigillito to this more general context.