The main conjecture on basic Euler systems over number fields

Let $K$ be a number field, let $\mathcal{A}^{\rm s}_K$ be the specified ideal in the relevant group ring, let ${\rm ES}_K$ be the module of Euler systems of rank $r_K$ for $\mathbb{G}_m$ over $K$, and let ${\rm ES}^{\rm b}_K=\operatorname{im}(\Theta_K^{\rm s})$ be the module of basic Euler systems arising from the canonical map $\Theta_K^{\rm s}$. Main conjecture on basic Euler systems. One has

$$\mathcal{A}^{\rm s}_K\cdot {\rm ES}_K\subseteq \mathcal{A}^{\rm s}_K\cdot {\rm ES}^{\rm b}_K.$$

Modulo minor technical issues concerning torsion, this predicts that all Euler systems in ${\rm ES}_K$ arise via the elementary construction giving ${\rm ES}^{\rm b}_K$.