Nontriviality conjecture for the ideals associated with frame parameters

Let $p\geq 5$ be prime, let $i\in\mathbb N_0$, and let $\gamma\in\hat{H}_i$. The ideal $J_i(\gamma)$ is therefore defined for the parameter $\gamma$ in the frame construction.

Nontriviality conjecture. For every such $p$, $i$, and $\gamma$,

$$J_i(\gamma)\neq\{0\}.$$

This would imply that $L_i(\gamma)$ is always finite and hence nilpotent; together with the stated class bound, its class is at most $p-1$ when $i>p-1$.