The force-based QC eigenvector conditioning conjecture

Let $V$ denote the matrix of eigenvectors for the force-based QC operator $L^{{\rm qcf}}_F$. Assume $A_F>0$. Eigenvector conditioning conjecture. As $N\to\infty$,

$$\operatorname{cond}(V)=o\left(\log(N)\right).$$

The conjecture is motivated by numerical experiments on the eigenvector basis and concerns the efficiency and stability of GMRES-type iterative methods; no proof or disproof is given in the paper.