Winding-number conjecture for type II runaways under diagonal Toeplitz perturbations

Let $T$ be the Toeplitz matrix and let $A_{jj}$ denote the matrix with its $(j,j)$ entry selected. Consider the rank-one deformation

$$T(\sigma)=T+\sigma A_{jj}.$$

Let the first-order perturbation correction be the corresponding collection of complex corrections, and let a Runaway type II be a runaway eigenvalue of this deformation. Winding-number conjecture. The number of Runaways type II in $T(\sigma)$ is equal to the winding number of the first-order perturbation correction about any point in the interior of the convex hull of the first-order corrections. This is an empirically observed relation for the rank-one deformation; the paper states that no proof is known and illustrates it numerically.