Serre's p-adic Stark conjecture at $s=1$

Let $E/\mathbb Q$ be the number field and $\bar G$ its relevant Galois group, let $E_0=\{x\in E:\operatorname{Tr}_{E/\mathbb Q}(x)=0\}$, and let $\mu_\infty$ and $\mu_p$ be the archimedean and $p$-adic regulator isomorphisms induced by logarithms on units. For $\rho\in R_{\mathbb C_p}(\bar G)$, define the leading term

$$L_{p,S}^*(1,\rho):=\lim_{s\to1}(s-1)^{\langle\rho,1\rangle}L_{p,S}(s,\rho).$$

Serre's p-adic Stark conjecture. The value $L_{p,S}^*(1,\rho)$ is the leading term of $L_{p,S}(s,\rho)$ at $s=1$, and for every isomorphism of $\mathbb Q[\bar G]$-modules $g:E_0\to\mathcal O_E^\times\otimes_{\mathbb Z}\mathbb Q$,

$$\frac{L_{p,S}^*(1,\rho)}{\text{det}_{\mathbb C_p}((\mathbb C_p\otimes_{\mathbb Q_p}\mu_p)\circ(\mathbb C_p\otimes_{\mathbb Q}g))^\rho}=\frac{L_S^*(1,\rho)}{\text{det}_{\mathbb C}(\mu_\infty\circ(\mathbb C\otimes_{\mathbb Q}g))^\rho}.$$

This compares the $p$-adic and complex leading terms through regulator determinants and is a $p$-adic analogue of Stark-type special-value conjectures; the assertion is open in the generality stated.