The product-formula conjecture for the quasi-eigenfunction at $\alpha=-t^{1/2}$ and $\alpha=t$

Let $F(\alpha)$ be the quasi-eigenfunction in the homogeneous specialization $s_1=\cdots=s_n=1$, and let $(a;q)_\infty$ denote the $q$-shifted factorial. Define $\prod_{1\leq i<j\leq n\atop\mathrm{step}\ 2}f_{ij}=f_{13}f_{15}\cdots f_{24}f_{26}\cdots$.

Product-formula conjecture. The following identities hold:

$$F(-t^{1/2})=\prod_{1\leq i<j\leq n}(1-\zeta_j/\zeta_i)\frac{(-qt^{-1/2}\zeta_j/\zeta_i;q)_\infty}{(-t^{1/2}\zeta_j/\zeta_i;q)_\infty},$$ $$F(t)=\prod_{1\leq i<j\leq n\atop\mathrm{step}\ 2}(1-\zeta_j/\zeta_i)\frac{(qt^{-1}\zeta_j/\zeta_i;q)_\infty}{(t\zeta_j/\zeta_i;q)_\infty}.$$

The identities are verified for $n=3$ under the conjectural formula for $F(\alpha)$ and checked using partial $n=4$ data. Their validity for general $n$ is left open.