A Bloch–Kato-type formula for critical Asai motives

Let $\mathcal{M}=\mathcal{M}(\pi,r^{\pm}\otimes\Psi)$ be critical, with Betti and de Rham realizations $H_{\rm B}(\mathcal{M})$ and $H_{\rm dR}(\mathcal{M})$, and let $T_{\mathfrak p}$ be the chosen $G_{K^+}$-invariant lattice in its $\mathfrak p$-adic realization. Let $\Omega$ be the corresponding Deligne period. For a finite set of primes $\Sigma^+$ containing every $v$ at which $\pi_v$ or $K/K^+$ is ramified, but not containing $p$, write $T^*_{\mathfrak p}=\operatorname{Hom}_{\mathcal O}(T_{\mathfrak p},\mathcal O)$ with the dual $G_{K^+}$-action. Conjecture 4.1. One has

$$\operatorname{ord}_{\mathfrak p}\left(\frac{L^{\Sigma^+}(1,\pi,r^{\pm}\otimes\Psi)}{\Omega}\right)=\operatorname{ord}_{\mathfrak p}\left(\frac{\#H^1_{\Sigma^+}(K^+,T^*_{\mathfrak p}\otimes(E/\mathcal O))}{\#H^0(K^+,T^*_{\mathfrak p}\otimes(E/\mathcal O))}\right),$$

where $\#$ denotes a Fitting ideal. This is the expected relation between the algebraic part of a critical Asai $L$-value and the corresponding Selmer group, in the spirit of the Bloch–Kato conjecture. The source does not provide evidence resolving this formulation.