Andrews–Dixit–Schultz–Yee oddness conjecture for a double Lambert series

Let

$$Y(q):=\sum_{n=1}^\infty\sum_{m=1}^\infty\frac{(-1)^m q^{2nm+m}}{(1-q^{2m-1})(1+q^n)}.$$

Here $q$ is a complex number with $|q|<1$, so the double Lambert series defines an analytic function in its domain of convergence. Andrews–Dixit–Schultz–Yee conjecture. The function $Y(q)$ is odd:

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

The conjecture was proposed as an alternative route to proving the congruences for the overpartition function $\overline{p}_\omega(n)$ at arguments $4n+3$ and $8n+6$. Its resolution is not specified in the supplied source material.