A coefficient identity involving a double Lambert series

Let $r$ be a positive integer, and let $[q^k]F(q)$ denote the coefficient of $q^k$ in $F(q)$. Coefficient conjecture. Then

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

This is a proposed coefficient relation between two Lambert-series expressions and is presented as the second conjecture concluding the section. The source gives no resolution or supporting result sufficient to classify it otherwise.