Vlasenko–Zwegers conjectural identities for the rank two Example 10 Nahm sums

Let $q$ be a formal variable, and let $J_{a,b}$ and $J_a$ denote the Jacobi-product notation used in the source. Vlasenko–Zwegers' conjecture. The following identities are conjectured:

$$\sum_{i,j\geq 0} \frac{q^{2i^2+2ij+2j^2-2i-j}}{(q^3;q^3)_i(q^3;q^3)_j}=\frac{1}{J_3}\left(2J_{18,45}+qJ_{12,45}+q^4J_{3,45}\right),$$ $$\sum_{i,j\geq 0} \frac{q^{2i^2+2ij+2j^2}}{(q^3;q^3)_i(q^3;q^3)_j}=\frac{1}{J_3}\left(J_{21,45}-q^3J_{6,45}+2q^2J_{9,45}\right).$$

Equivalently, the source also gives the corresponding bilateral theta-series expressions preceding these product forms. The identities concern modular rank-two Nahm sums; the supplied context says their modularity was confirmed, but describes the identities themselves as conjectural and does not provide a complete proof.