Conjecture on the integer algebraic spectrum of the spin chain

Let $\mathsf{H}$ be the spin-chain Hamiltonian, and let the algebraic eigenvectors and algebraic energies be those defined in the preceding construction. For $M\teq 1$, the first three algebraic levels are the three lowest levels of the chain, while the fourth algebraic energy is $\mathsf{E}=3$.

Integer-spectrum conjecture. The algebraic eigenvectors span the three lowest levels of the spin chain. When $M\teq1$, the fourth algebraic energy, $\mathsf{E}=3$, is the fifth lowest level, and its whole eigenspace is spanned by states of the specified form. Moreover,

$$\operatorname{spec}(\mathsf{H})\cap\mathbb{Z}=\{0,1,2,3\}.$$

This conjecture concerns how the explicitly constructed algebraic eigenvalues sit inside the full spectrum of the spin chain. The supplied text gives numerical corroboration but no proof of the asserted spectral ordering or of the characterization of all integer eigenvalues, so the claim remains open.