Finiteness conjecture for nonreduced Hessenberg NRS-matrices

Let $\Omega$ be an arbitrary Hessenberg type, and let $NRS(\Omega)$ denote the set of NRS-matrices of type $\Omega$. Call a matrix $\varsigma$-reduced according to the paper's reduction procedure. Finiteness conjecture. All but a finite number of NRS-matrices of type $\Omega$ are $\varsigma$-reduced. The theorem proved earlier establishes only finiteness along each NRS-ray; this conjecture asks for finiteness of the exceptional $\varsigma$-nonreduced matrices for every Hessenberg type, and the source presents it as an open problem.