Equivariant Artin conjecture for the smoothed zeta element

Assume the smoothed equivariant main conjecture without assuming uniqueness, and let $\zeta_S^T\in K_1(\mathcal Q(\mathcal G))$ satisfy

$$\operatorname{nr}(\zeta_S^T)=\Phi_S^T,\qquad \partial(\zeta_S^T)=[Y_S^T].$$

Let $\mathfrak M$ be a $\Lambda(\Gamma_0)$-order in $\mathcal Q(\mathcal G)$ containing $\Lambda(\mathcal G)$.

Equivariant Artin conjecture for the smoothed zeta element. The element $\zeta_S^T$ lies in the image of the natural map

$$\mathfrak M\cap\mathcal Q(\mathcal G)^\times\longrightarrow K_1(\mathcal Q(\mathcal G)).$$

This is an integrality conjecture refining the smoothed equivariant main conjecture. The source gives no resolution and explicitly does not assume uniqueness.