Integral polynomial degree for singlet components

In the trigonometric case $y=0$, choose the singlet components to be coprime polynomials in $x$ and normalise their leading terms as specified in the source. Let $N$ be written as either $N=2n+1$ or $N=2n$, and let $d_N$ denote the maximal degree of the distinguished singlet component. The degree conjecture. With this normalisation, all components are polynomials in $x$ with integer coefficients when

$$d_N=\begin{cases}n^2,&N=2n+1,\ (n-1),&N=2n.\end{cases}$$

The claim refines the observed asymptotic normalisation of the zero-energy singlet.