Bloch–Kato conjecture for the Asai L-function

Let $K/\mathbf{Q}$ be the quadratic extension, let $\pi$ be the automorphic representation and let $q>2k$ be a prime at which neither $\pi$ nor $K/\mathbf{Q}$ is ramified. Let $T_{\mathfrak{q}}^{(-1)^k}(k)^*(1)$ denote the dual of the chosen $\mathfrak{q}$-adic lattice in the Tate-twisted Asai representation, and let $\Omega$ be the corresponding Deligne period. For a finite set $\Sigma^+$ of primes containing every prime $p$ at which $\pi_p$ or $K/\mathbf{Q}$ is ramified, but not $q$, write $L^{\Sigma^+}(1,\pi,r^{(-1)^k})$ for the imprimitive Asai $L$-value and $\operatorname{ord}_{\mathfrak{q}}$ for the $\mathfrak{q}$-adic valuation. Bloch–Kato conjecture. One has

$$\operatorname{ord}_{\mathfrak{q}}\left(\frac{L^{\Sigma^+}(1,\pi,r^{(-1)^k})}{\Omega}\right)=\operatorname{ord}_{\mathfrak{q}}\left(\frac{\# H^1_{\Sigma^+}(\mathbf{Q},T_{\mathfrak{q}}^{(-1)^k}(k)^*(1)\otimes(E_{\mathfrak{q}}/\mathcal{O}_{\mathfrak{q}}))}{\# H^0(\mathbf{Q},T_{\mathfrak{q}}^{(-1)^k}(k)^*(1)\otimes(E_{\mathfrak{q}}/\mathcal{O}_{\mathfrak{q}}))}\right),$$

where $T_{\mathfrak{q}}^*={\rm Hom}_{\mathcal{O}_{\mathfrak{q}}}(T_{\mathfrak{q}},\mathcal{O}_{\mathfrak{q}})$ has the dual $G_{\mathbf{Q}}$-action and $\#$ denotes a Fitting ideal. This is the predicted relation between the special value of the Asai $L$-function and the Selmer-group arithmetic of the associated motive; the source gives no evidence here that the conjecture has been resolved.