Raising and lowering symmetry operators of the periodic Motzkin chain

Let $s_i^\pm$ be local spin-$1$ operators at site $i$, let $I$ be the identity operator, and let $\mathrm{H}^{\text{periodic}}$ be the periodic Motzkin-chain Hamiltonian. Define

\Sigma^\pm=\sum_{\substack{r_1,\dots,r_N\in\{-2,-1,0,1,2\}\r_1+\dots+r_N=\pm1}}s_1^{r_1}\cdots s_N^{r_N},

where $s_i^0\equiv I$, $s_i^{\pm1}\equiv s_i^\pm$, and $s_i^{\pm2}\equiv(s_i^\pm)^2$.

Raising and lowering operator conjecture. There exist raising and lowering operators $\Sigma^\pm$ satisfying

$$\Sigma^\pm\lvert v_{S^z}\rangle=\begin{cases}c_\pm(S^z)\lvert v_{S^z\pm1}\rangle,&S^z\ne\pm N,\\0,&S^z=\pm N,\end{cases}$$

with $c_\pm(S^z)\ne0$, and commuting with the Hamiltonian,

$$[\Sigma^\pm,\mathrm{H}^{\text{periodic}}]=0.$$

They are given by the displayed sum, equivalently by the residue formula in the source.

These operators would provide non-local symmetries connecting the conjectured ground states. The source reports verification through small system sizes but gives no proof or resolution.