Equivariant p-adic Artin conjecture for graduated orders

Let $p$ be an odd rational prime, let $K$ be a totally real number field, and let $\mathcal L/K$ be an admissible one-dimensional $p$-adic Lie extension. Write $\mathcal G=\operatorname{Gal}(\mathcal L/K)\simeq H\rtimes\Gamma$, where $H$ is finite and $\Gamma\simeq\mathbb Z_p$. Choose $n_0\gg0$ such that $\Gamma_0=\Gamma^{p^{n_0}}$ is central in $\mathcal G$. Let $S$ contain all places ramifying in $\mathcal L/K$ and all infinite places, let $T$ be a non-empty finite set of places disjoint from $S$, and let $\Phi_S^T$ be the associated smoothed equivariant $p$-adic Artin $L$-function. If $\mathfrak M$ is a $\Lambda(\Gamma_0)$-order in $\mathcal Q(\mathcal G)$ containing $\Lambda(\mathcal G)$, then the equivariant $p$-adic Artin conjecture asserts that $\Phi_S^T$ lies in the image of

$$\mathfrak M\cap \mathcal Q(\mathcal G)^\times\longrightarrow K_1(\mathcal Q(\mathcal G))\xrightarrow{\operatorname{nr}}\mathfrak z(\mathcal Q(\mathcal G))^\times,$$

where the first map sends an invertible element to the class of the corresponding $1\times1$ matrix. This conjecture is an equivariant analogue of the $p$-adic Artin conjectures of Greenberg, proved by Wiles and by Ritter--Weiss; the source records that the equivariant analogue was proposed previously, but gives no resolution of the graduated-order formulation.