The identity relating the double Lambert series to $D_2(q)-D_1(q)$

Let $Y(q)$ be the double Lambert series defined by

$$Y(q):=\sum_{m,n\geqslant 1} \frac{(-q)^{2mn+m}}{(1+q^m)(1-q^{2m-1})}, \qquad |q|<1,$$

and let $D_1(q)$ and $D_2(q)$ denote the functions defined earlier in the paper. In particular, the identity is considered in the paper's notation.

Identity conjecture.

$$Y(q)=D_2(q)-D_1(q).$$

The paper presents this as a conjecture in its future-work section and says that proving it by a more elementary method would simplify the proof of the preceding conjecture. Because the supplied context does not define $D_1(q)$ and $D_2(q)$, the precise formulation should be checked against the full paper.