Hall–Littlewood polynomial identity for the generalized Rogers–Ramanujan series

Let $m,n\geq 1$. Let $h_{\lambda}^{(m)}(w,z;q)$ denote the modified Rogers–Szegő factor defined earlier, let $P'_{\lambda}$ be the modified Hall–Littlewood polynomial, and let $F_{m,n}(w,z;q)$ be the Rogers–Ramanujan/Nahm–Zagier-type $q$-series defined in the paper. Generalized Rogers–Ramanujan identity. Specializing $x=q^{1/2}(u,u^{-1},u,\dots)$, with $n$ terms, in the left-hand side of the preceding identity yields

$$\sum_{\substack{\lambda\\[1.5pt] \lambda_1\leq 2m}} q^{|\lambda|/2}h_{\lambda}^{(m)}(w,z;q) P'_{\lambda}\big(\underbrace{u,u^{-1},u,\dots}_{n\ \text{terms}};q\big)=F_{m,n}(w,z;q).$$

This conjectural identity is intended to provide Hall–Littlewood polynomial expressions for important Rogers–Ramanujan-, Andrews–Gordon-, Bressoud-, and generalized Göllnitz–Gordon-type series; its general validity is not established in the supplied text.