The equality conjecture for hypergeometric logarithmic quotients

Let $n\ne2$ and let $a_i,b_i\in\mathbb{Q}$ satisfy $0<a_i,b_i<1$ for $i=1,2,\ldots,n$. Let $F(a\mid z)$ be the generalized hypergeometric solution and let $G(a\mid z)$ be its first logarithmic solution, with analogous notation for $b$. The equality conjecture for logarithmic quotients. The formal power series identity

$$\frac{G(b_1,b_2,\ldots,b_n\mid z)}{F(b_1,b_2,\ldots,b_n\mid z)}=\frac{G(a_1,a_2,\ldots,a_n\mid z)}{F(a_1,a_2,\ldots,a_n\mid z)}$$

holds if and only if

$$\{b_1,b_2,\ldots,b_n\}=\{a_1,a_2,\ldots,a_n\}.$$

The conjecture is used to characterize the parameter sets giving equal logarithmic quotients and to complete the converse direction of the integrality criterion. It is false over $\mathbb{C}$, since the associated variety has extra components; the remaining arithmetic assertion is that these extra components have no nontrivial $\mathbb{Q}$-rational points.