The universal formula for the quantum toroidal central coefficient

Let $N$, $L$, and $n$ be positive integers, and let $\gamma_n(q)$ be the scalar defined by

$$[B_n,B_{-n}]|M\rangle=\gamma_n(q)|M\rangle.$$

For $N=1$, $L=1$, or $n=1,2$, the preceding proposition gives

$$\gamma_n(q)=n\frac{1-q^{2Nn}}{1-q^{2n}}\frac{1-q^{-2Ln}}{1-q^{-2n}}.$$

Universal formula conjecture. The formula

$$\gamma_n(q)=n\frac{1-q^{2Nn}}{1-q^{2n}}\frac{1-q^{-2Ln}}{1-q^{-2n}}$$

is valid for all positive integers $N$, $L$, and $n$.

The claim extends the formula proved in the special cases $N=1$, $L=1$, and $n=1,2$; the source does not provide a proof for arbitrary positive $N$, $L$, and $n$.