Parity and defectivity conjecture for collisions of runaway eigenvalues

Let $T$ be the Toeplitz matrix, let $A_{jj}$ denote the indicated diagonal rank-one perturbation, and let $V$ denote the perturbation appearing in the source. Consider the deformations

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

At an eigenvalue collision, compare the geometric multiplicity with the algebraic multiplicity; the matrix is defective when the geometric multiplicity is smaller than the algebraic multiplicity. Defectivity conjecture. For $j$ even, the matrix $T(\sigma)=T+\sigma A_{jj}$ is defective when Runaways type I eigenvalues collide and become degenerate. When $j$ is odd, or when $T(\sigma)=T+\sigma V$, the matrix has the same geometric multiplicity as algebraic multiplicity at the moment of collision. The source presents this as an observed parity-dependent distinction in the collision behavior of type I runaway eigenvalues; no proof or resolution is supplied.