Unbounded-root conjecture for degenerate Lamé operators

Let $\mathfrak d(z)$ be a degenerate Lamé operator, meaning that its leading coefficient satisfies $\deg Q_k<k+r$, and let $N_0$ be any positive integer. Unbounded-root conjecture. The union of all roots of the Van Vleck polynomials $V$ and Stieltjes polynomials $S$, taken over all solutions with $\deg S\geq N_0$, is unbounded. Thus, this property is proposed as a key distinction between non-degenerate and degenerate Lamé operators. The supplied text gives no resolution and explicitly presents the claim as a conjecture motivated by earlier results.