The Asai recognition conjecture for special hypergeometric motives

Let $\boldsymbol\alpha$ be a set of parameters from Table 1 and let $\boldsymbol\beta=\{1,1,1,1,1\}$. For a hypergeometric motive $H(\boldsymbol\alpha,\boldsymbol\beta\mid z)$, write $L_p(H(\boldsymbol\alpha,\boldsymbol\beta\mid z),T)$ for its local $L$-factor at a good prime $p$. Let $F$ be a real quadratic field, let $f$ be a Hilbert cusp form over $F$ of weight $(2,4)$, and let $L_p(f,T,\operatorname{Asai},\varepsilon)$ and $L(f,s,\operatorname{Asai},\varepsilon)$ denote the Asai local and global $L$-functions, respectively. Asai recognition conjecture. There exist quadratic Dirichlet characters $\chi,\varepsilon$ and such a Hilbert cusp form $f$ over a real quadratic field $F$ such that, for all good primes $p$,

$$L_p(H(\boldsymbol\alpha,\boldsymbol\beta\mid z),T)\overset?=(1-\chi(p)p^2T)L_p(f,T/p,\operatorname{Asai},\varepsilon).$$

In particular,

$$L(H(\boldsymbol\alpha,\boldsymbol\beta\mid z),s)\overset?=L(s-2,\chi)L(f,s+1,\operatorname{Asai},\varepsilon).$$

This is the paper's main conjecture, proposing that the relevant special hypergeometric motives are recognized through Asai $L$-functions of Hilbert cusp forms, with an additional quadratic Dirichlet factor. The conjecture is presented without a resolution in the source.