Augmented down-up algebra grading conjecture for $\hat{\mathbb T}$

Let $q$ be the parameter and let $e,D$ be the parameters occurring in the defining relations of the algebra $\hat{\mathbb T}$, generated by $R,L,K,K^{-1}$, with central elements $C_1,C_2$. An algebra grading is a direct-sum decomposition

$$\hat{\mathbb T}=\bigoplus_{n\in\mathbb Z}\hat{\mathbb T}_n.$$

The grading conjecture. The degree-zero component has basis

$$K^hC_1^iC_2^j\qquad h\in\mathbb Z,\quad i,j\in\mathbb N,$$

the component of degree $n\geq1$ has basis

$$R^nK^hC_1^iC_2^j\qquad h\in\mathbb Z,\quad i,j\in\mathbb N,$$

and the component of degree $-n$ has basis

$$L^nK^hC_1^iC_2^j\qquad h\in\mathbb Z,\quad i,j\in\mathbb N.$$

The paper notes that this should be verifiable using the Bergman diamond lemma; no proof is supplied there.