The irrationality conjecture for a ratio of hypergeometric values

Let

$$F(a,b)={}_3F_2\left(\begin{matrix}a,\,b,\,1\\1,\,1\end{matrix};1\right)$$

be the hypergeometric value used in the paper. Irrationality conjecture.

$$\frac{F(5/12,7/12)}{F(7/12,17/12)}\notin\mathbb{Q}(\sqrt{3}).$$

This mild irrationality assumption is introduced for class 8, where the relevant even twist is not an inner twist and the second part of Shimura's result cannot be applied. It is used to handle the corresponding transformation of hypergeometric series; the source provides no resolution of the conjecture.