Sum rules for the three singlets in the cyclic model

For odd $N=2n+1$, consider the cyclic spin-chain Hamiltonian with parameters $\lambda,\zeta$ and its three translation-invariant zero-energy singlets $\Psi_N^a$, indexed by $a=0,\pm$, with components $\psi_\mu^a$ in spin configurations $\mu$. Let $\nu_1(\mu)$ be the number of spins equal to $1$ in $\mu$. The cyclic singlet sum-rule conjecture. With the stated normalisation, the square norm of each singlet factorises into two polynomials in $\zeta,\lambda$ with positive integer coefficients. Along $\lambda=\zeta$,

$$\lVert\Psi_N^a(\zeta)\rVert^2=3^{-2n}\left(\sum_\mu\zeta^{\nu_1(\mu)}\psi_\mu^a\right)\left(\sum_\mu\zeta^{-\nu_1(\mu)}\psi_\mu^a\right),\qquad a=0,\pm.$$

For $\Psi_N^0$, along the same line,

$$\sum_\mu\zeta^{\nu_1(\mu)}\psi_\mu^0=3^{2n}\zeta^{2n+1}\psi_{11\dots1}^0.$$

The source presents these as open questions supported by exact diagonalisation through $N=7$.