Higher-spin XXZ identification at the root of unity

Let $h'$ be the modified local Hamiltonian and let $h_{\mathrm{XXZ}_{\ell/2}}$ be the spin-$\ell/2$ higher-spin XXZ Hamiltonian density defined from the quadratic Casimir construction. The higher-spin XXZ identification. One has

$$h'=h_{\mathrm{XXZ}_{\ell/2}}$$

at $q=\exp\left(\text{i}\pi/(\ell+2)\right)$. This was verified explicitly through $\ell=6$, while the general assertion is presented as a statement beyond the cases checked directly.