A Rogers–Ramanujan type double-sum identity with index $(1,3)$

For $|q|<1$, let $(a;q)_n$ denote the $q$-Pochhammer symbol, defined by $(a;q)_n=\prod_{k=0}^{n-1}(1-aq^k)$ and $(a;q)_\infty=\prod_{k=0}^{\infty}(1-aq^k)$; for multiple parameters, write $(a_1,\ldots,a_r;q)_\infty=\prod_{s=1}^r(a_s;q)_\infty$. The $(1,3)$ Rogers–Ramanujan type identity conjecture.

$$\sum_{i,j\geq 0}\frac{q^{i^2+3j^2-3ij+j}}{(q;q)_i(q^3;q^3)_j}=\frac{(q^6;q^9)_\infty}{(q,q^2,q^2,q^4,q^5,q^5;q^6)_\infty}.$$

This identity is proposed as a new companion to an earlier conjectured identity for Rogers–Ramanujan type identities with index $(1,3)$; its status is not resolved in the supplied source.