Conjugate Bailey-pair conjecture for affine A-type string functions

Let $\ell\in\mathbb{Z}_{+}$, $k\in Q\cap P_{+}$, and $\eta\in\mathcal{P}$ satisfy $\eta_1\leq n-1$ and $|\eta|\equiv\ell\pmod n$. Let $\Lambda\in P_{+}^{N}$ satisfy $\ell\bar{\Lambda}_1-\bar{\Lambda}\in Q$. Conjugate Bailey-pair conjecture. The quantities

$$\gamma_k=\mathcal{C}_{\ell\bar{\Lambda}_1-\pi(k),\Lambda}(q),\qquad \delta_\eta=X_{\eta,\Lambda,N\Lambda_0}(q)$$

form an $A_{n-1}$ conjugate Bailey pair relative to $q^{\ell}$. This extends the preceding special case derived from known theorems; no general proof or resolution is given in the supplied text.