Generalized configuration-sum fermionic formula

Let $\eta\in\mathcal{P}$ satisfy $\eta_1\leq n-1$. For $s,u\in\{0,\dots,N\}$ and $r,v\in\{0,\dots,n-1\}$, set $\Lambda=(N-s)\Lambda_r+s\Lambda_{r+1}$ and $\Lambda'=(N-u)\Lambda_0+u\Lambda_{n-v}$, with $|\eta|\bar{\Lambda}_1-\bar{\Lambda}+\bar{\Lambda}'\in Q$. Define $F_{\eta,\Lambda,\Lambda'}(q)$ by the multiple sum and constraints in the source. Generalized configuration-sum conjecture. Under these conditions,

$$X_{\eta,\Lambda,\Lambda'}(q)=F_{\eta,\Lambda,\Lambda'}(q).$$

The source presents this as a generalization of an earlier theorem and uses it to derive further identities; no proof or resolution is given in the supplied text.