Fredkin-chain conjecture on the anti-adjoint representation of the raising and lowering operators

Consider the periodic Fredkin chain with $N$ sites, total-spin operators $\mathrm{S}^\pm$, and the raising and lowering operators $\Sigma^\pm$. Define the anti-adjoint action by

$$(\operatorname*{\mathrm{aad}} a)\, b \equiv \{a,b\}=ab+ba.$$

Anti-adjoint representation conjecture. There exist coefficients $\gamma_k$ such that

$$\Sigma^\pm =\sum_{k=1}^{\lceil N/2\rceil } \gamma_k \left(\operatorname*{\mathrm{aad}} \mathrm{S}^\pm \operatorname*{\mathrm{aad}} \mathrm{S}^\mp\right)^{k-1} \mathrm{S}^\pm.$$

The claim gives a representation of the nonlocal operators $\Sigma^\pm$ in terms of the non-diagonal components of the total spin operator. It was inferred from finite-size calculations and is one of the paper's conjectural descriptions of the Fredkin-chain symmetry operators.