The charge-refined q-series identity for the principal subspace of type B

Let $q$, $y_1$, and $y_2$ be formal variables, and let $(q)_n$ denote the finite $q$-Pochhammer symbol for $n\geq 0$. The variables $r_1,r_2,r_3,n_1,n_2,n_3,n_4,n_5$ range independently over nonnegative integers.

Charge-refined q-series conjecture. The following identity holds:

$$\sum_{\substack{r_1,r_2,r_3 \geq 0}} y_1^{r_1} y_2^{2r_3+r_2} \frac{q^{r_1^2+(r_2+r_3)^2+r_3^2-r_1(r_2+2 r_3)}}{(q)_{r_1}(q)_{r_2}(q)_{r_3}}= \sum_{\substack{n_1,n_2,n_3,n_4,n_{5} \geq 0}}$$ $$y_1^{n_1+n_2+n_4} y_2^{2n_1+2n_3+n_4+n_5} \frac{q^{n_1^2+n_2^2+(n_3+n_5)^2+n_4^2+n_3^2+(2 n_3+n_5)(n_1)+n_4(n_1+n_2)+n_3 n_4}}{(q)_{n_1}(q)_{n_2}(q)_{n_3}(q)_{n_4}(q)_{n_5}}.$$

This stronger charge-refined identity is intended to establish the expected equality of Hilbert-series and character formulas for the principal subspace of the level-one $so(5)$ representation; it was checked computationally only to finite order in the source, so its general validity remains open.