Realization and arithmetic compatibility conjecture for rank 2 hypergeometric motives

Let $K$ be the base field of the hypergeometric data $\boldsymbol\alpha,\boldsymbol\beta$. For a parameter $z$, write $\mathcal H(\boldsymbol\alpha,\boldsymbol\beta\mid z)$ for the associated hypergeometric motive, and let $S_g$ denote the set of primes of good reduction. For a finite extension $L$ of $K$, consider a generic specialization $z\in L$.

Realization and compatibility conjecture. The following assertions hold:

1. The motive $\mathcal H(\boldsymbol\alpha,\boldsymbol\beta\mid z)$ has a realization $\mathcal H(z)$ over $K(z)$.
2. The representation of the motive restricted to $\operatorname{Gal}(\overline{K(z)}/\overline K(z))$ is isomorphic to the geometric Galois representation.
3. The specialization $\mathcal H(z)$ for generic $z\in L$ has coefficient field $K$.
4. The primes $\mathfrak p\in S_g$ of $L$ are of good reduction, and the trace of Frobenius on $\mathcal H(z)$ is given by the finite hypergeometric sum $H_{\mathfrak p}(\boldsymbol\alpha,\boldsymbol\beta\mid z)$.

These assertions are expected from the conjugation behavior of hypergeometric motives and are supported by numerical evidence. They formulate the conjectural realization, compatibility, coefficient-field, and Frobenius-trace properties for the rank-$2$ hypergeometric motives considered in the paper.