Hall–Littlewood and Rogers–Szegő polynomial identity

Let $M=(M_1,\dots,M_m)\in\mathbb Z_{+}^m$ and set $m_0(\lambda):=\infty$. For a partition $\lambda$, write $\lambda'$ for its conjugate, $m_j(\lambda)$ for the multiplicity of the part $j$, $\lambda_{\operatorname{o}}$ for the partition formed by the odd-sized parts of $\lambda$, $\ell(\lambda_{\operatorname{o}})$ for its length, $P'_{\lambda}(x;q)$ for the modified Hall–Littlewood polynomial, and $h_{\lambda_{\operatorname{o}}}(w/z;q)$ for the corresponding Rogers–Szegő polynomial. Let $f_{r,s}^{(2)}(x;q)$ denote the functions defined earlier in the paper. Hall–Littlewood–Rogers–Szegő conjecture. For $M=(M_1,\dots,M_m)\in\mathbb Z_{+}^m$, one has

$$\sum_{\substack{\lambda\\[1.5pt] \lambda_1\leq 2m}} z^{\ell(\lambda_{\operatorname{o}})} P'_{\lambda}(x;q) h_{\lambda_{\operatorname{o}}}(w/z;q) \prod_{\ell=1}^m (wz)^{M_{\ell}-\lambda'_{2\ell-1}} \genfrac{[}{]}{0pt}{}{m_{2\ell-2}(\lambda)}{M_{\ell}-\lambda'_{2\ell-1}} =\sum \prod_{i=1}^n \big( {-}q^{1-r^{(1)}_i}w/x_i,-q^{1-r^{(1)}_i}z/x_i\big)_{r^{(1)}_i} \prod_{\ell=1}^m f_{r^{(\ell)},r^{(\ell+1)}}^{(2)}(x;q),$$

where the sum on the right is over $r^{(1)},\dots,r^{(m)}\in\mathbb Z_{+}^n$ such that $|r^{(\ell)}|=M_{\ell}$, and $r^{(m+1)}:=0$. This is presented as a general conjectural identity and is proved only in special cases related to the affine Lie algebras $\mathrm C_n^{(1)}$ and $\mathrm A_{2n}^{(2)}$; the general identity remains open.