Non-specialized Capparelli–Meurman–Primc–Primc conjecture for C_n^(1)

Let $k_0,\ldots,k_n$ be non-negative integers. Let $\mathbb{Z}_{\mathcal{S}}^+$ be the coloured-part set, let $\Omega$ be the set of fictitious parts, and let $\mathcal{P}^{k_0,\ldots,k_n}_{\mathcal{S}}$ consist of partitions whose frequencies, together with fictitious frequencies $f_{\omega_i}=k_i$, satisfy

$$f_{e_0}+\ldots+f_{e_{2n}}\leq k_0+\ldots+k_n$$

for every path $(e_0,\ldots,e_{2n})$ in $\Omega\sqcup\mathbb{Z}_{\mathcal{S}}^+$. Define $q=e^{-\delta}$, $c_i=e^{\frac{\alpha_n}{2}+\sum_{u=i}^{n-1}\alpha_u}$, and $c_{\overline{i}}=c_i^{-1}$ for $i\in\{1,\ldots,n\}$. Non-specialized CMPP conjecture. One has

$$\sum_{\pi\in \mathcal{P}^{k_0,\ldots,k_n}_{\mathcal{S}}} C(\pi)q^{|\pi|}=e^{-k_0\Lambda_0-\cdots-k_n\Lambda_n}\operatorname{ch}(L(k_0\Lambda_0+\cdots+k_n\Lambda_n)).$$

This is proposed as a non-specialized generalization of the principal character conjecture; the supplied text gives no resolution status.