Factorisation of the supersymmetry singlet norm

For twist angle $\phi=\pi$, write the supersymmetry singlet as $|\Psi\rangle=\sum_\mu\psi_\mu|\mu\rangle$, with the components chosen as coprime polynomials in $x,y$. The norm-factorisation conjecture. The square norm $\lVert\Psi\rVert^2$ factorises into two polynomials in $x,y$; when $N$ is odd, one factor is the component $\psi_{11\cdots1}$. This is an observed sum rule for the singlet states.