Cao–Wang's unresolved modular rank four Nahm sum identity

Let $q$ be a formal variable. For $n\geq 0$, write $(q^2;q^2)_n=\prod_{r=1}^{n}(1-q^{2r})$, and let $J_m=(q^m;q^m)_\infty$. Then

Cao–Wang's conjecture.

$$\sum_{i,j,k,l\geq 0}\frac{q^{2i^2+j^2+2k^2+l^2-2ik-2il-jl+2i-j-2k}}{(q^2;q^2)_i(q^2;q^2)_j(q^2;q^2)_k(q^2;q^2)_l}=9\frac{J_3^3J_6^2}{J_1J_2^4}.$$

This is one of the conjectural modular rank four Nahm sum identities proposed by Cao and Wang. The paper proves five related identities but states that it is currently unable to resolve this one.