Gross's interpolation conjecture for $p$-adic Artin $L$-functions

Fix an odd prime $p$, a field isomorphism $\iota:\mathbb{C}\simeq\mathbb{C}_p$, a character $\psi\in\operatorname{Irr}_{\mathbb{C}_p}(\mathcal{G})$, and a finite set of places $S$ of $K$ containing all archimedean places and all places ramified in $\mathcal{L}/K$. Let $L_{p,S}(s,\psi)$ be the $S$-truncated $p$-adic Artin $L$-function, let $L_S(s,\cdot)$ be the $S$-truncated Artin $L$-function, and write $\psi^{\iota}=\iota^{-1}\circ\psi$; let $\omega$ denote the relevant cyclotomic character. Gross's interpolation conjecture. For each $\psi\in\operatorname{Irr}_{\mathbb{C}_p}(\mathcal{G})$, one has

$$L_{p,S}(0,\psi)=\iota\left(L_S(0,(\psi\omega^{-1})^{\iota})\right).$$

The usual interpolation property is known for linear characters, and extends to characters of arbitrary degree when the interpolation point is $1-r$ with $r\geq 2$. The conjecture concerns the remaining case $r=1$, where the standard argument does not apply.