Fredkin-chain conjecture on annihilation of non-cyclic invariant eigenstates

Let $\Sigma^\pm$ be the raising and lowering operators for the periodic Fredkin chain, let $C$ be the cyclic shift operator, and let $\mathrm{H}$ be its Hamiltonian. An eigenstate is non-cyclic invariant when its cyclic-shift eigenvalue satisfies $C\ne 1$. Annihilation conjecture. The operators $\Sigma^\pm$ annihilate all non-cyclic invariant eigenstates of the Hamiltonian. This conjecture extends the observed annihilation of the state $\left\lvert v_0^- \right\rangle$ for even $N$ and concerns the action of the proposed symmetry generators outside the cyclic-invariant sector; it was formulated from computations for chains with $N\leq 10$.