The refined p-adic Beilinson conjecture for motives

Let $k/\mathbb Q$ be finite Galois with group $G$, let $E/\mathbb Q$ be finite, let $\pi\in E[G]$ be an idempotent, and let $\psi_\pi$ and $\chi_\pi$ be the associated representation and character. Let $M_\pi^E$ and its archimedean and $p$-adic discriminants and regulators be as defined in the source. Assume

$$\dim_E(E[G]\pi)=\dim_E(\pi K_{2n-1}(k))$$

for some $n\geq2$.

The refined p-adic Beilinson conjecture. There are $e(n,M_\pi^E),e_p(n,M_\pi^E)\in(E\otimes_{\mathbb Q}\mathbb Q)^*$ such that the archimedean and $p$-adic regulator formulas hold, with $e_p(n,M_\pi^E)=e(n,M_\pi^E)$; additionally, $L_p(n,\chi_\pi\otimes\omega_p^{1-n},\mathbb Q)$ and $R_{n,p}(M_\pi^E)$ are units in the indicated tensor products.

This refines the number-field formula by incorporating idempotent components and comparison of complex and $p$-adic periods. The source gives no resolution.