The preconditioned force-based QC spectral equivalence conjecture

Let $N\geq4$, let $1\leq K\leq N-2$, and let $F>0$. Let ${L}^{-1}$ be the inverse of the discrete Laplacian operator $L$, and consider the preconditioned operators ${L}^{-1}L^{{\rm qcf}}_F$ and ${L}^{-1}L^{{\rm qnl}}_F$. Preconditioned spectral equivalence conjecture. The operator ${L}^{-1}L^{{\rm qcf}}_F$ is diagonalizable and has the same spectrum as ${L}^{-1}L^{{\rm qnl}}_F$. This conjecture extends the unpreconditioned spectral observation to the generalized eigenvalue problem used to study preconditioning in the $\mathcal{U}^{1,2}$ setting; the paper supplies numerical evidence but no proof.