The symmetrically preconditioned force-based QC eigenvector conditioning conjecture

Let ${\widetilde W}$ denote the matrix of eigenvectors for the symmetrically preconditioned force-based QC operator ${L}^{-1/2}L^{{\rm qcf}}_F{L}^{-1/2}$. Assume $A_F>0$. Symmetric preconditioned conditioning conjecture. As $N\to\infty$,

$$\operatorname{cond}({\widetilde W})=O\left(N^{3}\right).$$

This conjecture concerns the eigenvector conditioning of the symmetric formulation of the preconditioned problem; it is supported by numerical experiments in the paper, while the asymptotic bound remains unproved there.