The rational-factor conjecture for the quasi-eigenfunction at $\alpha=-q^{\ell}t^{1/2}$

Let $F(\alpha)$ be the homogeneous quasi-eigenfunction in $n\geq2$ variables. Let $\ell$ be a positive integer, let $J=\{(\sigma_1,\ldots,\sigma_\ell)\mid\sigma_i=\pm\}$, and let $G_\ell(\zeta)=(\gamma_{\ell,\sigma,\sigma'}(\zeta))_{\sigma,\sigma'\in J}$ be the recursively defined matrix of rational functions. For $\sigma\in J$, set

$$\mu_\sigma=\prod_{i=1}^{\ell}\left(q^{(i-1)/2}t^{1/4}\right)^{\sigma_i}.$$

Rational-factor conjecture. At $\alpha=-q^\ell t^{1/2}$,

$$\begin{aligned} F(-q^\ell t^{1/2})={}&\left(\sum_{\sigma\in J}\mu_\sigma\right)^{-n}\prod_{1\leq i<j\leq n}(1-\zeta_j/\zeta_i)\frac{(-qt^{-1/2}\zeta_j/\zeta_i;q)_\infty}{(-t^{1/2}\zeta_j/\zeta_i;q)_\infty}\\ &\times\sum_{\sigma_1,\ldots,\sigma_n\in J}\prod_{i=1}^n\mu_{\sigma_i}\prod_{1\leq i<j\leq n}\gamma_{\ell,\sigma_i,\sigma_j}(\zeta_j/\zeta_i). \end{aligned}$$

This extends the observed factorization pattern to positive integers $\ell$. It is presented as an observation based on explicit calculations, and the general formula remains conjectural.