Amdeberhan–Andrews–Ballantine coefficient identity for a double Lambert series

Let $r$ be a positive integer. The notation $[q^n]A(q)$ denotes the coefficient of $q^n$ in the series $A(q)$. Amdeberhan–Andrews–Ballantine coefficient identity.

$$[q^{2r}]\sum_{m,n\geq1}\frac{q^{2mn}}{(1+q^{2n-1})(1-q^{2m-1})}=[q^{2r}]\sum_{n\geq1}\frac{(n-1)q^n}{1+q^{2n-1}}.$$

The identity was proposed as one of two conjectures used in the study of generalized and double Lambert series, and as a route toward a conjecture of Andrews, Dixit, Schultz and Yee. The source states that it has since been proved by the first author and Tang.