The quasi-eigenfunction transformation conjecture

Set $s_1=\cdots=s_n=1$ and write $I(\alpha)=I(\alpha;1,\ldots,1,q,t)$. Let $F(\alpha)$ be the quasi-eigenfunction defined by $I(\alpha q^{-1}t)F(\alpha)=F(\alpha q^{-1}t)$ and the stated initial condition at $\alpha=t^{1/2}$.

Transformation conjecture. The function $F(\alpha)$ also satisfies

$$I(\alpha^{-1}q)\cdot F(\alpha)=F(\alpha q^{-1}t).$$

The first transformation property defines $F(\alpha)$ through iteration and analytic continuation, and the second is presented as an expected additional symmetry. It is checked only in limited expansions in the supplied text.