The proposed Rogers–Ramanujan type character identity for type $A^{(2)}_{6n+3}$

Let $n\text{ be an integer}$ with $n\geq 1$, and let $b$ satisfy $0\leq b\leq 2n$. For nonnegative integers $i,j,k,\ell_1,\dots,\ell_{2n}$, define

$$L_k=\ell_k+\dots+\ell_{2n}\qquad (1\leq k\leq 2n).$$

Define

$$Q_n(i,j,k,\ell_1,\dots,\ell_{2n})=n(i+3j+3k)^2+2(L_1^2+\dots+L_{2n}^2)+2(i+3j+3k)(L_1+\dots+L_{2n})+2iL_{2n},$$

and

$$R_{n,b}(i,j,k,\ell_1,\dots,\ell_{2n})=(b+2)i+(3b+2)j+(3b+4)k+2(L_{2n+1-b}+L_{2n+2-b}+\dots+L_{2n}).$$

Rogers–Ramanujan type character identity. For $n\geq 1$ and $0\leq b\leq 2n$, the identity

$$\sum_{i,j,k,\ell_1,\dots,\ell_{2n}\geq 0}\frac{q^{Q_n(i,j,k,\ell_1,\dots,\ell_{2n})+R_{n,b}(i,j,k,\ell_1,\dots,\ell_{2n})}}{(q^2;q^2)_i(q^6;q^6)_j(q^6;q^6)_k(q^2;q^2)_{\ell_1}\cdots(q^2;q^2)_{\ell_{2n}}}=\chi_{A^{(2)}_{6n+3}}(\Lambda'_{n+2+b})$$

holds. This is a Rogers–Ramanujan type identity whose right-hand side is the principal character of a level $2$ standard module of type $A^{(2)}_{6n+3}$, extending the family of identities discussed in the paper. The supplied passage does not establish whether this proposed identity is proved or remains conjectural.