Algebra grading conjecture for generalized uniform-poset algebras

Fix distinct integers $s,t$, a finite set $\mathbb I\subseteq\mathbb Z\setminus\{s,t\}$, nonzero scalars $\gamma_i\in\mathbb F$ for $i\in\mathbb I$, and

$$\phi(K)=\sum_{i\in\mathbb I}\gamma_iK^i.$$

Let $\alpha_s,\alpha_t$ be central and let $S$ be the algebra with

$$RL=\alpha_sK^s+\alpha_tK^t+\phi(K),$$ $$LR=\alpha_sq^{-2s}K^s+\alpha_tq^{-2t}K^t+\phi(q^{-2}K).$$

The generalized grading conjecture. The corresponding algebra $S$ has a direct-sum algebra grading

$$S=\bigoplus_{n\in\mathbb Z}S_n,$$

where $S_0$ has basis $K^h\alpha_s^i\alpha_t^j$, $S_n$ has basis $R^nK^h\alpha_s^i\alpha_t^j$ for $n\geq1$, and $S_{-n}$ has basis $L^nK^h\alpha_s^i\alpha_t^j$ for $n\geq1$, in every case with $h\in\mathbb Z$ and $i,j\in\mathbb N$. The source gives no proof or resolution.