The even and odd Nahm sum identities for $\hat{Z}_e(q)$ and $\hat{Z}_o(q)$

Let $q$ be a formal variable, let $(q^2;q^2)_n=\prod_{r=1}^{n}(1-q^{2r})$ for $n\geq 0$, let $J_m=(q^m;q^m)_\infty$, and define the bilateral theta-type series $f(x,y)=\sum_{r\in\mathbb Z}x^{r(r+1)/2}y^{r(r-1)/2}$. Define

$$\hat{Z}_e(q)=\sum_{a,j,s\geq 0}\frac{q^{a^2-aj+j^2-j+2s^2+2s-2sj}}{(q^2;q^2)_a(q^2;q^2)_j(q^2;q^2)_{2s}},$$

and

$$\hat{Z}_o(q)=\sum_{a,j,s\geq 0}\frac{q^{a^2-aj+j^2-j+2s^2+4s+1-(2s+1)j}}{(q^2;q^2)_a(q^2;q^2)_j(q^2;q^2)_{2s+1}}.$$

The even and odd Nahm sum conjecture.

$$\hat{Z}_e(q)=3\frac{J_3^3}{J_1J_2^2J_6}f(q^4,q^8),$$

and

$$\hat{Z}_o(q)=3\frac{J_3^3}{J_1J_2^2J_6}f(q^2,q^{10}).$$

If established, this conjecture implies the Cao–Wang identity above. It is presented as a conjecture in the paper and is connected there to a conjecture of Shi and Wang; no resolution is supplied in the given text.