Gross's second-part conjecture on leading coefficients

Let $\psi\in\operatorname{Irr}_{\mathbb C_p}(\mathcal G)$, choose $\iota:\mathbb C\simeq\mathbb C_p$, and let $r_S(\psi)$ be the corresponding complex order of vanishing. Let $L_S^*(0,(\psi\omega^{-1})^\iota)$ be the leading coefficient of the complex $S$-truncated Artin $L$-function at $s=0$, and let $R_{p,S}^{(\iota)}(\psi)$ be the explicit $p$-adic regulator defined in the paper. Gross's second-part conjecture. One has

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

This is the leading-coefficient refinement of the order-of-vanishing assertion and relates special values through a $p$-adic regulator. It remains open in general.