The modular-product q-series identity of type B

Let $(q)_n$ denote the finite $q$-Pochhammer symbol for $n\geq 0$, and let $(a_1,\ldots,a_k;q)_\infty$ denote the corresponding infinite multiple $q$-Pochhammer product. The summation variables $r_1,r_2,r_3$ range over nonnegative integers.

Modular-product q-series conjecture. After setting $y_1=y_2=1$ in the charge-refined identity, the following identity holds:

$$\sum_{\substack{r_1,r_2,r_3 \geq 0}} \frac{q^{r_1^2+(r_2+r_3)^2+r_3^2-r_1(r_2+2 r_3)}}{(q)_{r_1}(q)_{r_2}(q)_{r_3}}=\frac{(-q;q)_\infty (-q;q^2)^2_\infty}{(q,q^4;q^5)_\infty}.$$

The identity is presented as an elegant specialization with a modular product side, but the source gives no proof or resolution; it is related to the preceding charge-refined conjecture and is conjectured to arise from a more complicated quantum-dilogarithm relation.