A q-product identity for F_0^star(q) and G_0^star(q)

Let $q$ be a formal variable, let $\omega$ be a primitive third root of unity, and define

$$F_0^{\star}(q):=\sum_{a,b\geq 0}\frac{q^{a^2-3ab+3b^2+a-2b}}{(q^3;q^3)_b}\begin{bmatrix}3b-a-1\a\end{bmatrix}_q,$$ $$G_0^{\star}(q):=\sum_{a,b\geq 0}\frac{q^{a^2-3ab+3b^2-2a+4b+1}}{(q^3;q^3)_b}\begin{bmatrix}3b-a+2\a\end{bmatrix}_q.$$

Here $(a;Q)_\infty=\prod_{j\geq0}(1-aQ^j)$, multiple parameters are interpreted multiplicatively, and $\begin{bmatrix}n\k\end{bmatrix}_q$ denotes the $q$-binomial coefficient. The q-product identity for F_0^star(q) and G_0^star(q). The following identity is expected to be true:

$$F_0^{\star}(q)+\omega^2G_0^{\star}(q)=-\omega\cdot \frac{(q^{30};q^{45})_{\infty}(\omega^2 q^3;q^3)_{\infty}(\omega q^{5},\omega q^{7},\omega q^{11},\omega q^{13},\omega q^{14};q^{15})_{\infty}}{(q,q^{4},q^{10};q^{15})_{\infty}(q^{18},q^{27},q^{33},q^{42};q^{45})_{\infty}}.$$

This is the second of two conjectures introduced in the section and is related to earlier identities of Warnaar, Uncu, and Zudilin. Its status is not resolved in the supplied source.